Modeling with linear equations example 2

WEBVTT
Kind: captions
Language: en

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In this video, I want to do a
couple more word problems

00:00:02.820 --> 00:00:05.060
dealing with graphs of lines.

00:00:05.060 --> 00:00:07.010
So here we have a
gym is offering

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a deal to new members.

00:00:08.900 --> 00:00:12.040
Customers can sign up by paying
a registration fee of

00:00:12.040 --> 00:00:22.690
$200 and then a monthly
fee of $39.

00:00:22.690 --> 00:00:23.940
This is registration.

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How much will this membership
cost a member by

00:00:31.120 --> 00:00:32.890
the end of the year?

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So let's figure out an equation
that determines how

00:00:35.710 --> 00:00:39.470
much total we will pay.

00:00:39.470 --> 00:00:42.680
p is equal to the amount that
we're going to pay in total

00:00:42.680 --> 00:00:44.230
for our membership.

00:00:44.230 --> 00:00:48.300
So no matter how many months we
use the gym, just to start

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using the gym, we have to pay
$200 in registration.

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I'll just write-- everything
we'll assume is in dollars

00:00:56.100 --> 00:00:57.830
--so I'll write $200.

00:00:57.830 --> 00:01:00.800
And then we're going to have
to pay $39 for every month

00:01:00.800 --> 00:01:01.400
we're there.

00:01:01.400 --> 00:01:04.200
So then we're going to take
the number of months we're

00:01:04.200 --> 00:01:08.430
there and multiply
that times 39.

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Notice if we stay there 1 month,
we'll have to pay 1

00:01:11.130 --> 00:01:12.550
month times $39.

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And we would have already paid
the $200 registration fee.

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So it'll be $239.

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If we stay 2 months, we pay the
$200 registration fee and

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then we pay 39 times 2 months,
which is what?

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Like 78 or something.

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So it would be $278.

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So just to tie this altogether
with linear equations and

00:01:30.630 --> 00:01:33.430
graphs of them, let's
graph this relation.

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Remember the graph of
a line can be y is

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equal to mx plus b.

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That's one of the forms. So to
put this line in this form or

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this equation in this form, we
can just rearrange the 39m and

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the 200 and you get p is
equal to 39m plus 200.

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So what's the slope and what's
the y-intercept?

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You might get confused and say,
hey there's an x there

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and a y, but now you're doing
it with p's and m's.

00:02:01.670 --> 00:02:09.699
Just remember this is the
independent variable and this

00:02:09.699 --> 00:02:10.780
is the dependent variable.

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Here this is the independent
variable.

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How many months?

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You pick a number of months
and I'll tell you what the

00:02:15.210 --> 00:02:17.490
total cost of your membership
is going to be.

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It's the same thing.

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This is like the
x right there.

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This is like the y
just like that.

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Just using our pattern match,
this right here is the-- we

00:02:27.570 --> 00:02:30.810
could say it's the vertical
intercept or the p-intercept

00:02:30.810 --> 00:02:35.030
or the-- I'm tempted to call
it the y-intercept.

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But we're really intersecting
the p-axis instead of the

00:02:37.970 --> 00:02:39.430
y-axis there.

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This right here is our slope.

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So let's graph this function.

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I won't do it too accurately.

00:02:47.710 --> 00:02:49.550
I just want to do a hand
drawn graph just

00:02:49.550 --> 00:02:50.820
to give you an idea.

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We could just stay in
the first quadrant.

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We're not going to stay negative
months and the gym is

00:02:54.860 --> 00:02:56.840
never going to pay us money.

00:02:56.840 --> 00:02:58.852
So right off the bat,
we're going to have

00:02:58.852 --> 00:03:05.610
to pay the gym $200.

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$200 for 0 months.

00:03:08.040 --> 00:03:09.380
Then for every month, we're
going to have to

00:03:09.380 --> 00:03:11.450
spend an extra $39.

00:03:11.450 --> 00:03:13.380
So the slope is 39.

00:03:13.380 --> 00:03:16.800
Let's say this is 1
month right there.

00:03:16.800 --> 00:03:18.050
This is in months.

00:03:20.250 --> 00:03:23.380
And this axis is price,
the p-axis.

00:03:23.380 --> 00:03:26.040
So this is like the
p-intercept or the

00:03:26.040 --> 00:03:27.190
y-intercept.

00:03:27.190 --> 00:03:29.940
So after 1 month, how much are
we going to have to pay?

00:03:29.940 --> 00:03:33.000
Well our slope is 39, so if we
move 1 month forward, we're

00:03:33.000 --> 00:03:34.730
going to go up by 39.

00:03:34.730 --> 00:03:38.630
So this will, right here,
that will be 239.

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If we go another month,
it'll be 278.

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This is kind of a weird labeling
of an axis, but I

00:03:46.880 --> 00:03:48.190
think you get the idea.

00:03:48.190 --> 00:03:50.800
So the graph of how much it'll
cost us as per month will look

00:03:50.800 --> 00:03:54.630
something like this.

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So they say, how much will
a membership cost by

00:03:56.510 --> 00:03:57.340
the end of the year?

00:03:57.340 --> 00:03:58.060
12 months.

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We would have to go 2, 3, all
the way out to 12 months,

00:04:01.840 --> 00:04:02.750
which might be here.

00:04:02.750 --> 00:04:04.565
So then our graph is going
to be out here someplace.

00:04:04.565 --> 00:04:07.650
But we could just figure
it out algebraically.

00:04:07.650 --> 00:04:10.830
At the end of the year,
m will be equal to 12.

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When m is equal to 12, how
much is our membership?

00:04:14.090 --> 00:04:16.870
The price of our membership
is going to be our $200

00:04:16.870 --> 00:04:20.829
membership fee plus 39
times the number of

00:04:20.829 --> 00:04:23.110
months, times 12.

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What's 39 times 12?

00:04:26.760 --> 00:04:30.500
2 times 9 is 18.

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2 times 3 is 6 plus 1 is 70.

00:04:33.930 --> 00:04:34.680
I have a 0.

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1 times 9 is 9.

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1 times 3-- we want
to ignore this.

00:04:38.680 --> 00:04:40.910
1 times 3 is 3.

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So we have 8.

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7 plus 9 is 16.

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1 plus 3 is 4.

00:04:47.370 --> 00:04:51.230
So the price of our membership
is $200 plus 39 times 12,

00:04:51.230 --> 00:04:54.120
which is $468.

00:04:54.120 --> 00:04:58.240
So it's equal to $668 at
the end of our year.

00:04:58.240 --> 00:05:00.200
So if you went all the way out
to 12, you would have to plot

00:05:00.200 --> 00:05:03.360
668 someplace here on
our line, if we just

00:05:03.360 --> 00:05:05.290
kept going out there.

00:05:05.290 --> 00:05:08.310
Let's do one more of these.

00:05:08.310 --> 00:05:13.070
Bobby and Petra are running a
lemonade stand and they charge

00:05:13.070 --> 00:05:15.430
$0.45 for each glass
of lemonade.

00:05:21.780 --> 00:05:26.650
In order to break even,
they must make $25.

00:05:26.650 --> 00:05:31.410
How many glasses of lemonade
must they sell to break even?

00:05:31.410 --> 00:05:34.980
So let me just do
it with y and x.

00:05:34.980 --> 00:05:37.875
y is equal to the amount
they make.

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Not max-- the amount
they make.

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Let x is equal to the number
of glasses they sell.

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What is y as a function of x?

00:05:59.180 --> 00:06:04.040
So y is equal to-- well for
every glass they sell, they

00:06:04.040 --> 00:06:09.090
get $0.45 --so it's equal
to $0.45 times

00:06:09.090 --> 00:06:10.800
the number of glasses.

00:06:10.800 --> 00:06:13.760
There's not any kind of minimum
fee that they need to

00:06:13.760 --> 00:06:16.440
charge or they don't say any
kind of minimum cost that they

00:06:16.440 --> 00:06:18.430
have to spend to
run this place.

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How much in order to
break even for

00:06:21.480 --> 00:06:25.170
each a glass of lemonade?

00:06:25.170 --> 00:06:28.020
They need to make $25.

00:06:28.020 --> 00:06:33.300
So in order to break even,
they must make $25.

00:06:33.300 --> 00:06:37.570
So how many glasses of lemonade
do they need to sell?

00:06:37.570 --> 00:06:40.360
y needs to be equal to $25.

00:06:40.360 --> 00:06:42.530
How many glasses do
they need to sell?

00:06:42.530 --> 00:06:44.230
Well you just set
this equation.

00:06:44.230 --> 00:06:55.310
You say 0.45x has to
be equal to 25.

00:06:55.310 --> 00:07:03.860
We can divide both
sides by 0.45.

00:07:03.860 --> 00:07:05.440
On the left-hand side, you're
just left with an x.

00:07:05.440 --> 00:07:14.875
You get x is equal to-- What
is 25 divided by 0.45?

00:07:14.875 --> 00:07:21.050
It is equal to-- They would
have to sell exactly 55.55

00:07:21.050 --> 00:07:24.150
glasses or 56 if I round.

00:07:24.150 --> 00:07:28.310
55.5 repeating glasses.

00:07:28.310 --> 00:07:30.520
But you can't sell half a glass
or we're assuming you

00:07:30.520 --> 00:07:32.340
can't sell half a glass.

00:07:32.340 --> 00:07:36.372
So the answer to that, they
must sell 56 glasses.

00:07:36.372 --> 00:07:38.960
Because you can't sell half
a glass, I'm assuming.

00:07:38.960 --> 00:07:42.250
So they need to sell 56
glasses to break even.

00:07:42.250 --> 00:07:44.530
Just to graph this.

00:07:44.530 --> 00:07:46.930
Once again, we'll hang out in
the first quadrant because

00:07:46.930 --> 00:07:49.620
everything is going
to be positive.

00:07:49.620 --> 00:07:52.750
Every glass they make $0.45.

00:07:52.750 --> 00:07:54.570
Let's say that they sell--
So this is the

00:07:54.570 --> 00:07:56.390
number of glasses, x.

00:07:56.390 --> 00:07:58.010
This is how much they make.

00:07:58.010 --> 00:08:04.500
Let me go by increments of 5.

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5, 10, 15, 20, 25.

00:08:09.730 --> 00:08:11.810
Actually I need to go by even
larger increments to get to

00:08:11.810 --> 00:08:13.530
the point that we're
talking about.

00:08:13.530 --> 00:08:14.520
Let me go increments of 10.

00:08:14.520 --> 00:08:21.270
10, 20, 30, 40, 50, 60.

00:08:21.270 --> 00:08:23.320
So that's the number
of glasses.

00:08:23.320 --> 00:08:25.700
When they sell 0 glasses,
they make $0.

00:08:25.700 --> 00:08:27.940
That's their y-intercept.

00:08:27.940 --> 00:08:29.740
y is equal to 0.

00:08:29.740 --> 00:08:33.260
When they sell 10 glasses,
they make $4.50.

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So this is 4.50.

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This is 9.

00:08:37.640 --> 00:08:41.230
Actually let me just
do it like this.

00:08:41.230 --> 00:08:50.500
Let me just mark only
the multiples of 9.

00:08:50.500 --> 00:08:59.610
Let's say 9, 18, 27, 35.

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When they sell 10 glasses,
they're going to make $4.50.

00:09:03.700 --> 00:09:05.170
10 times 0.45.

00:09:05.170 --> 00:09:06.540
That's right there.

00:09:06.540 --> 00:09:09.240
20 glasses, they're going
to make $9.00.

00:09:09.240 --> 00:09:10.670
We can keep going there.

00:09:10.670 --> 00:09:12.940
40 glasses, they're
going to make $18.

00:09:12.940 --> 00:09:13.810
You see their slope.

00:09:13.810 --> 00:09:17.220
When you move 10, you're going
to have to go up 4.50.

00:09:17.220 --> 00:09:20.860
This graph is going to look
something like that.

00:09:20.860 --> 00:09:22.520
Should be a straight line.

00:09:22.520 --> 00:09:24.430
Then if you want to see their
break even, their break even

00:09:24.430 --> 00:09:29.125
has to be $25, which is
right about here.

00:09:32.170 --> 00:09:35.130
Their break even is $25
right around there.

00:09:38.220 --> 00:09:41.600
Let me draw the line a little
bit better than that.

00:09:41.600 --> 00:09:45.020
The line is going to
look like this.

00:09:45.020 --> 00:09:52.240
If the break even is $25-- it
would be right there --you see

00:09:52.240 --> 00:09:56.720
that they have to sell
about 56 glasses.

00:09:56.720 --> 00:09:59.930
Obviously the way I drew this
isn't the super neatly drawn

00:09:59.930 --> 00:10:04.010
graph, but hopefully it gives
you the general idea.


Evaluating with function notation

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.460
In this video, I want
to do a few examples

00:00:02.460 --> 00:00:03.800
dealing with functions.

00:00:03.800 --> 00:00:06.570
Functions tend to be something
that a lot of students find

00:00:06.570 --> 00:00:09.230
difficult, but I think if you
really get what we're talking

00:00:09.230 --> 00:00:11.070
about, you'll see that it's
actually a pretty

00:00:11.070 --> 00:00:12.240
straightforward idea.

00:00:12.240 --> 00:00:13.710
And you sometimes wonder,
well what was all

00:00:13.710 --> 00:00:14.880
of the hubbub about?

00:00:14.880 --> 00:00:16.720
All a function is, is an

00:00:16.720 --> 00:00:19.830
association between two variables.

00:00:19.830 --> 00:00:25.540
So if we say that y is equal to
a function of x, all that

00:00:25.540 --> 00:00:28.260
means is, you give me an x.

00:00:28.260 --> 00:00:31.660
You can imagine this function
is kind of eating up this x.

00:00:31.660 --> 00:00:34.190
You pop an x into
this function.

00:00:34.190 --> 00:00:36.480
This function is just
a set of rules.

00:00:36.480 --> 00:00:39.150
It's going to say, oh,
with that x, I

00:00:39.150 --> 00:00:41.230
associate some value y.

00:00:41.230 --> 00:00:42.945
You can imagine it is
some type of a box.

00:00:45.900 --> 00:00:47.990
That is a function.

00:00:47.990 --> 00:00:53.830
When I give it some number
x, it'll give me some

00:00:53.830 --> 00:00:56.990
other number y.

00:00:56.990 --> 00:00:58.160
This might seem a
little abstract.

00:00:58.160 --> 00:00:59.360
What are these x's and y's?

00:00:59.360 --> 00:01:02.830
Maybe I have a function-- let
me make it like this.

00:01:02.830 --> 00:01:04.190
Let's say I have a function
definition

00:01:04.190 --> 00:01:05.720
that looks like this.

00:01:05.720 --> 00:01:11.770
For any x you give me, I'm going
to produce 1 if x is

00:01:11.770 --> 00:01:14.440
equal to-- I don't know-- 0.

00:01:14.440 --> 00:01:18.730
I'm going to produce 2
if x is equal to 1.

00:01:18.730 --> 00:01:21.320
And I'm going to produce
3 otherwise.

00:01:24.790 --> 00:01:28.720
So now we've defined what's
going on inside of the box.

00:01:28.720 --> 00:01:31.630
So let's draw the
box around it.

00:01:31.630 --> 00:01:33.650
This is our box.

00:01:33.650 --> 00:01:35.940
This is just an arbitrary
function definition, but

00:01:35.940 --> 00:01:37.760
hopefully it'll help you
understand what's actually

00:01:37.760 --> 00:01:40.070
going on with a function.

00:01:40.070 --> 00:01:47.500
So now if I make x is equal to--
if I pick x is equal to

00:01:47.500 --> 00:01:52.480
7, now what is f of x going
to be equal to?

00:01:52.480 --> 00:01:56.400
What is f of 7 going
to be equal to?

00:01:56.400 --> 00:01:58.020
So I take 7 into the box.

00:01:58.020 --> 00:01:59.700
You could view it as some
type of a computer.

00:01:59.700 --> 00:02:02.770
The computer looks at that x and
then looks at its rules.

00:02:02.770 --> 00:02:04.060
It says, OK. x is 7.

00:02:04.060 --> 00:02:06.270
Well x isn't 0. x isn't 1.

00:02:06.270 --> 00:02:08.229
I go to the otherwise
situation.

00:02:08.229 --> 00:02:10.100
So I'm going to pop out a 3.

00:02:10.100 --> 00:02:12.040
So f of 7 is equal to 3.

00:02:12.040 --> 00:02:15.320
So we write f of 7
is equal to 3.

00:02:15.320 --> 00:02:18.760
Where f is the name of this
function, this rule system, or

00:02:18.760 --> 00:02:21.310
this association, this
mapping, whatever you

00:02:21.310 --> 00:02:22.190
want to call it.

00:02:22.190 --> 00:02:24.350
When you give it a 7,
it'll produce a 3.

00:02:24.350 --> 00:02:27.460
When you give f a 7,
it'll produce a 3.

00:02:27.460 --> 00:02:31.240
What is f of 2?

00:02:31.240 --> 00:02:34.690
Well, that means instead of x
is equal to 7, I'm going to

00:02:34.690 --> 00:02:36.420
give it an x equal 2.

00:02:36.420 --> 00:02:38.550
Then the little computer inside
the function is going

00:02:38.550 --> 00:02:42.550
to say, OK, let's see,
when x is equal to 2.

00:02:42.550 --> 00:02:44.410
No, I'm still in the otherwise
situation.

00:02:44.410 --> 00:02:45.910
x isn't 0 or 1.

00:02:45.910 --> 00:02:50.800
So once again f of
x is equal to 3.

00:02:53.470 --> 00:02:56.970
So, this is f of 2 is
also equal to 3.

00:02:56.970 --> 00:03:03.200
Now what happens if x
is now equal to 1?

00:03:03.200 --> 00:03:05.100
Well then it's just going
to turn over this.

00:03:05.100 --> 00:03:07.990
So f of 1.

00:03:07.990 --> 00:03:10.080
It's going to look at its
rules right here.

00:03:10.080 --> 00:03:11.620
Oh look, x is equal to 1.

00:03:11.620 --> 00:03:13.350
I can use my rule right here.

00:03:13.350 --> 00:03:15.520
So when x is equal to
1, I spit out a 2.

00:03:15.520 --> 00:03:18.750
So f of 1 is going
to be equal to 2.

00:03:18.750 --> 00:03:22.290
I spit out f of 1, which is
equal to 2 in that situation.

00:03:22.290 --> 00:03:24.420
That's all a function is.

00:03:24.420 --> 00:03:29.120
Now, with that in mind, let's
do some of these example

00:03:29.120 --> 00:03:31.620
problems. They tell us for
each of the following

00:03:31.620 --> 00:03:35.010
functions, evaluate these
different functions-- these

00:03:35.010 --> 00:03:37.570
are the different boxes they've
created-- at these

00:03:37.570 --> 00:03:39.070
different points.

00:03:39.070 --> 00:03:42.800
Let's do part a first. They're
defining the box.

00:03:42.800 --> 00:03:47.880
f of x is equal to negative
2x plus 3.

00:03:47.880 --> 00:03:51.790
They want to know what happens
when f is equal to negative 3.

00:03:51.790 --> 00:03:54.300
Well f is equal to negative 3,
this is telling me what do I

00:03:54.300 --> 00:03:55.430
do with the x?

00:03:55.430 --> 00:03:57.110
What do I produce?

00:03:57.110 --> 00:04:00.060
Wherever I see an x, I replace
it with the negative 3.

00:04:00.060 --> 00:04:02.060
So it's going to be equal
to negative 2.

00:04:02.060 --> 00:04:04.780
Let me do it this way, so you
see exactly what I'm doing.

00:04:04.780 --> 00:04:06.520
That negative 3, I'll do
it in that bold color.

00:04:06.520 --> 00:04:13.130
It's negative 2 times
negative 3 plus 3.

00:04:13.130 --> 00:04:16.149
Notice wherever there was an
x, I put the negative 3.

00:04:16.149 --> 00:04:19.250
So I know what the black box
is going to produce.

00:04:19.250 --> 00:04:21.600
This is going to be equal to
negative 2 times negative 3 is

00:04:21.600 --> 00:04:25.640
6 plus 3, which is equal to 9.

00:04:25.640 --> 00:04:29.470
So f of negative 3
is equal to 9.

00:04:29.470 --> 00:04:32.130
What about f of 7?

00:04:32.130 --> 00:04:36.340
I'll do the same thing one more
time. f of-- I'll do 7 in

00:04:36.340 --> 00:04:43.120
yellow-- f of 7 is going to
be equal to negative 2

00:04:43.120 --> 00:04:47.650
times 7 plus 3.

00:04:50.480 --> 00:04:55.140
So this is equal to negative 14
plus 3, which is equal to

00:04:55.140 --> 00:04:57.260
negative 11.

00:04:57.260 --> 00:05:03.940
You put in-- let me make it very
clear-- you put in a 7

00:05:03.940 --> 00:05:11.060
into our function f here and it
will pop out a negative 11.

00:05:11.060 --> 00:05:13.310
That's what this just
told us right there.

00:05:13.310 --> 00:05:14.760
This is the rule.

00:05:14.760 --> 00:05:18.470
This is completely analogous
to what I did up here.

00:05:18.470 --> 00:05:20.980
This is the rule of
our function.

00:05:20.980 --> 00:05:24.430
Let's do the next two.

00:05:24.430 --> 00:05:25.200
I won't do part b.

00:05:25.200 --> 00:05:26.330
You can do part b for fun.

00:05:26.330 --> 00:05:29.650
I'll do part c after that, just
for the sake of time.

00:05:29.650 --> 00:05:32.540
Now we are at f of 0.

00:05:32.540 --> 00:05:33.810
Here I'll just do
it in one color.

00:05:33.810 --> 00:05:35.300
I think you're getting
the idea. f of 0.

00:05:35.300 --> 00:05:37.500
Wherever we see an
x, we put a 0.

00:05:37.500 --> 00:05:40.005
So negative 2 times 0 plus 3.

00:05:43.100 --> 00:05:44.345
Well, that's just
going to be a 0.

00:05:44.345 --> 00:05:47.300
So f of 0 is 3.

00:05:47.300 --> 00:05:49.000
Then one last one. f of z.

00:05:49.000 --> 00:05:51.720
They want to keep it
abstract for us.

00:05:51.720 --> 00:05:52.780
Here I'll color code it.

00:05:52.780 --> 00:05:55.800
So f of z.

00:05:55.800 --> 00:05:59.150
Let me make the z in
a different color.

00:05:59.150 --> 00:06:00.900
f of z.

00:06:00.900 --> 00:06:06.210
Everywhere that we saw
an x, we will now

00:06:06.210 --> 00:06:07.750
replace it with a z.

00:06:07.750 --> 00:06:09.240
Negative 2.

00:06:09.240 --> 00:06:12.040
Instead of an x, we're going
to put a z there.

00:06:12.040 --> 00:06:13.860
We're going to put an
orange z there.

00:06:13.860 --> 00:06:19.760
Negative 2 times z plus 3.

00:06:19.760 --> 00:06:24.330
And that's our answer. f of
z is negative 2z plus 3.

00:06:24.330 --> 00:06:28.110
If you imagine our box,
the function f.

00:06:28.110 --> 00:06:38.130
You put in a z, you are going to
get out a negative 2 times

00:06:38.130 --> 00:06:43.480
whatever that z is plus 3.

00:06:43.480 --> 00:06:44.520
That's all this is saying.

00:06:44.520 --> 00:06:47.830
It's a little bit more abstract,
but same exact idea.

00:06:47.830 --> 00:06:52.030
Now let's just do part c here.

00:06:52.030 --> 00:06:53.330
Let me clear this actually.

00:06:53.330 --> 00:06:55.820
I'm running out of
real estate.

00:06:55.820 --> 00:06:59.102
Let me clear all of
this business.

00:06:59.102 --> 00:07:02.910
Let me clear all of
this business.

00:07:02.910 --> 00:07:03.810
We can do part c.

00:07:03.810 --> 00:07:05.370
I'm skipping part b.

00:07:05.370 --> 00:07:07.710
You can work on that part.

00:07:07.710 --> 00:07:10.830
Part b.

00:07:10.830 --> 00:07:13.430
They tell us-- this is our
function definition.

00:07:13.430 --> 00:07:16.680
Sorry, I said I was
doing part c.

00:07:16.680 --> 00:07:18.610
This is our function
definition.

00:07:18.610 --> 00:07:26.300
f of x is equal to 5 times
2 minus x over 11.

00:07:26.300 --> 00:07:29.440
So let's apply these different
values of x, these different

00:07:29.440 --> 00:07:32.620
inputs into our function.

00:07:32.620 --> 00:07:39.900
So f of negative 3 is equal to
5 times 2 minus-- wherever we

00:07:39.900 --> 00:07:42.250
see an x, we put a negative 3.

00:07:42.250 --> 00:07:45.620
2 minus negative 3 over 11.

00:07:45.620 --> 00:07:48.700
This is equal to 2 plus 3.

00:07:48.700 --> 00:07:50.870
This is equal to 5.

00:07:50.870 --> 00:07:53.260
So you get 5 times 5 over 11.

00:07:53.260 --> 00:07:57.120
That's equal to 25/11.

00:07:57.120 --> 00:07:57.850
Let's do this one.

00:07:57.850 --> 00:07:59.990
F of 7.

00:07:59.990 --> 00:08:06.680
For this second function right
here, f of 7 is equal to 5

00:08:06.680 --> 00:08:11.160
times 2 minus-- now
for x we have a 7.

00:08:11.160 --> 00:08:14.360
2 minus 7 over 11.

00:08:14.360 --> 00:08:15.540
So what is this going
to be equal to?

00:08:15.540 --> 00:08:18.250
2 minus 7 is negative 5.

00:08:18.250 --> 00:08:23.780
5 times negative 5 is
negative 25/11.

00:08:23.780 --> 00:08:27.410
Then finally, well we have
two more. f of 0.

00:08:27.410 --> 00:08:35.000
That's equal to 5 times 2 minus
0 So this is just 2.

00:08:35.000 --> 00:08:36.130
5 times 2 is 10.

00:08:36.130 --> 00:08:38.850
So this is equal to 10/11.

00:08:38.850 --> 00:08:39.840
One more.

00:08:39.840 --> 00:08:42.059
f of z.

00:08:42.059 --> 00:08:43.299
Well every time we saw
an x, we're going to

00:08:43.299 --> 00:08:44.490
replace it with a z.

00:08:44.490 --> 00:08:49.960
It's equal to 5 times
2 minus z over 11.

00:08:49.960 --> 00:08:50.630
And that's our answer.

00:08:50.630 --> 00:08:51.910
We could distribute the 5.

00:08:51.910 --> 00:08:57.210
You could say this is the same
thing as 10 minus 5z over 11.

00:08:57.210 --> 00:09:00.260
We could even write it in
slope-intercept form.

00:09:00.260 --> 00:09:06.000
This is the same thing as
minus 5/11 z plus 10/11.

00:09:06.000 --> 00:09:06.990
These are all equivalent.

00:09:06.990 --> 00:09:10.430
But that is what f
of z is equal to.

00:09:10.430 --> 00:09:11.590
Now.

00:09:11.590 --> 00:09:15.510
A function, we said, if you give
me any x value, I will

00:09:15.510 --> 00:09:16.470
give you an output.

00:09:16.470 --> 00:09:19.120
I will give you an f of x.

00:09:19.120 --> 00:09:23.040
So if this is our function,
you give me an x, it will

00:09:23.040 --> 00:09:26.550
produce an f of x.

00:09:26.550 --> 00:09:29.680
It can only produce 1
f of x for any x.

00:09:29.680 --> 00:09:32.840
You can't have a function that
could produce two possible

00:09:32.840 --> 00:09:34.700
values for an x.

00:09:34.700 --> 00:09:37.540
So you can't have a function--
this would be an invalid

00:09:37.540 --> 00:09:42.790
function definition-- f
of x is equal to 3 if

00:09:42.790 --> 00:09:45.230
x is equal to 0.

00:09:45.230 --> 00:09:49.240
Or it could be equal to
4 if x is equal to 0.

00:09:49.240 --> 00:09:53.170
Because in this situation, we
don't know what f of 0 is.

00:09:53.170 --> 00:09:54.090
What it's going to be equal?

00:09:54.090 --> 00:09:56.330
It says if x is equal to 0, it
should be 3 or it could be--

00:09:56.330 --> 00:09:57.310
we don't know.

00:09:57.310 --> 00:09:57.830
We don't know.

00:09:57.830 --> 00:09:58.190
We don't know.

00:09:58.190 --> 00:10:01.550
This is not a function
even though it might

00:10:01.550 --> 00:10:02.800
have looked like one.

00:10:07.700 --> 00:10:12.250
So you can't have two f of
x values for one x value.

00:10:12.250 --> 00:10:16.020
So let's see which of these
graphs are functions.

00:10:16.020 --> 00:10:18.390
To figure that out, you could
say, look at any x value

00:10:18.390 --> 00:10:21.850
here-- pick any x value-- I have
exactly one f of x value.

00:10:21.850 --> 00:10:25.090
This is y is equal to
f of x right here.

00:10:25.090 --> 00:10:28.950
I have exactly only one--
at that x, that

00:10:28.950 --> 00:10:30.550
is my y value here.

00:10:30.550 --> 00:10:32.970
So you could have a vertical
line test, which says at any

00:10:32.970 --> 00:10:35.720
point if you draw a vertical
line-- notice a vertical line

00:10:35.720 --> 00:10:37.570
is for a certain x value.

00:10:37.570 --> 00:10:41.920
That shows that I only have
one y value at that point.

00:10:41.920 --> 00:10:43.630
So this is a valid function.

00:10:43.630 --> 00:10:46.240
Any time you draw a vertical
line, it will only intersect

00:10:46.240 --> 00:10:47.610
the graph once.

00:10:47.610 --> 00:10:50.410
So this is a valid function.

00:10:50.410 --> 00:10:52.220
Now what about this
one right here?

00:10:52.220 --> 00:10:53.960
I could draw a vertical
line, let's say, at

00:10:53.960 --> 00:10:55.230
that point right there.

00:10:55.230 --> 00:10:58.650
For that x, this relation
seems to have two

00:10:58.650 --> 00:11:00.860
possible f of x's.

00:11:00.860 --> 00:11:04.550
f of x could be that value or
f of x could be that value.

00:11:04.550 --> 00:11:05.270
Right?

00:11:05.270 --> 00:11:07.520
We're intersecting
the graph twice.

00:11:07.520 --> 00:11:08.840
So this is not a function.

00:11:08.840 --> 00:11:11.150
We're doing exactly what
I described here.

00:11:11.150 --> 00:11:15.090
For a certain x, we're
describing two possible y's

00:11:15.090 --> 00:11:16.800
that could be equal to f of x.

00:11:16.800 --> 00:11:19.220
So this is not a function.

00:11:19.220 --> 00:11:20.830
Over here, same thing.

00:11:20.830 --> 00:11:22.310
You draw a vertical
line right there.

00:11:22.310 --> 00:11:24.540
You're intersecting
the graph twice.

00:11:24.540 --> 00:11:26.000
This is not a function.

00:11:26.000 --> 00:11:30.590
You're defining two possible
y values for 1 x value.

00:11:30.590 --> 00:11:31.490
Let's go to this function.

00:11:31.490 --> 00:11:33.160
It's kind of a weird
looking function.

00:11:33.160 --> 00:11:34.750
Looks like a reversed
check mark.

00:11:34.750 --> 00:11:37.020
But any time you draw a vertical
line, you're only

00:11:37.020 --> 00:11:38.720
intersecting it once.

00:11:38.720 --> 00:11:40.420
So this is a valid function.

00:11:40.420 --> 00:11:43.470
For every x, you only have
one y associated.

00:11:43.470 --> 00:11:46.450
Or only one f of x associated
with it.

00:11:46.450 --> 00:11:48.960
Anyway, hopefully you
found that useful.


Direct variation models

WEBVTT
Kind: captions
Language: en

00:00:00.510 --> 00:00:03.780
Let's do some word problems that
are essentially dealing

00:00:03.780 --> 00:00:05.970
with slope of a line.

00:00:05.970 --> 00:00:08.250
You might see these being
referred to as direct

00:00:08.250 --> 00:00:11.280
variation models, because we're
going to model what's

00:00:11.280 --> 00:00:13.590
being described in
this problem.

00:00:13.590 --> 00:00:15.045
We're going to graph it out and
then hopefully we're going

00:00:15.045 --> 00:00:17.620
to be able actually answer
their question.

00:00:17.620 --> 00:00:18.540
Let's see what they're asking.

00:00:18.540 --> 00:00:23.750
The current standard for
low-flow showerheads is 2.5

00:00:23.750 --> 00:00:25.000
gallons per minute.

00:00:32.189 --> 00:00:35.390
Calculate how long it would
take to fill a 30 gallon

00:00:35.390 --> 00:00:39.180
bathtub using such a showerhead

00:00:39.180 --> 00:00:41.340
to supply the water.

00:00:41.340 --> 00:00:44.010
What we can do here is we can
set up a direct variation

00:00:44.010 --> 00:00:46.310
model, which sounds very fancy,
but it just says, OK,

00:00:46.310 --> 00:00:50.690
let's set up a little equation
that describes how many

00:00:50.690 --> 00:00:53.610
gallons we would have filled--
or how many gallons we would

00:00:53.610 --> 00:00:58.980
have used after a certain
number of minutes.

00:00:58.980 --> 00:01:05.600
So let's say that we have
gallons is equal to the rate

00:01:05.600 --> 00:01:06.780
at which we fill the gallons.

00:01:06.780 --> 00:01:17.460
So it's going to be 2.5 gallons
per minute times the

00:01:17.460 --> 00:01:18.710
number of minutes.

00:01:21.060 --> 00:01:24.380
I'll say m for minutes
and g for gallons.

00:01:24.380 --> 00:01:27.860
We have just set up our direct
variation model.

00:01:27.860 --> 00:01:28.780
Nothing fancier than that.

00:01:28.780 --> 00:01:31.360
We now have an equation that
describes, you give me the

00:01:31.360 --> 00:01:34.430
number of minutes, I'll multiply
that times 2.5

00:01:34.430 --> 00:01:37.550
because that's how quickly we're
filling the bathtub.

00:01:37.550 --> 00:01:41.460
So after 1 minute, 1 times
2.5, we have 2.5 gallons.

00:01:41.460 --> 00:01:45.220
After 2 minutes, we have 2 times
2.5, we have 5 gallons.

00:01:45.220 --> 00:01:46.330
So this is our model.

00:01:46.330 --> 00:01:48.350
And this is also a line.

00:01:48.350 --> 00:01:53.130
Remember the form of a line is
y is equal to mx plus b.

00:01:53.130 --> 00:01:54.510
Here we have no b.

00:01:54.510 --> 00:01:55.320
The b is gone.

00:01:55.320 --> 00:01:57.200
We just have an m times an x.

00:01:57.200 --> 00:02:00.250
The x now we're calling minutes
and the y we're now

00:02:00.250 --> 00:02:03.760
calling gallons and the
slope is now 2.5.

00:02:03.760 --> 00:02:07.670
Let's plot this before I even
answer their question.

00:02:07.670 --> 00:02:10.690
Instead of calling this the
x-axis-- remember x, the

00:02:10.690 --> 00:02:12.730
independent variable,
is now the minutes

00:02:12.730 --> 00:02:13.850
that we let it flow.

00:02:13.850 --> 00:02:16.470
So that's the m-axis
for minutes.

00:02:16.470 --> 00:02:19.070
And the vertical axis, instead
of calling it the y-axis, I'm

00:02:19.070 --> 00:02:21.260
going to call it the g-axis
for the number of gallons

00:02:21.260 --> 00:02:22.700
we've filled.

00:02:22.700 --> 00:02:24.866
I'm only going to deal in the
positive quadrant assuming we

00:02:24.866 --> 00:02:26.710
can only have positive
minutes.

00:02:26.710 --> 00:02:27.620
So what's going on here?

00:02:27.620 --> 00:02:29.820
We have a slope of 2.5.

00:02:29.820 --> 00:02:32.900
We could also write this as
gallons is equal to-- 2.5 is

00:02:32.900 --> 00:02:42.930
the same thing as 5/2 gallons
per minute times the minutes.

00:02:42.930 --> 00:02:45.030
So now we know our
slope is 5/2.

00:02:45.030 --> 00:02:48.280
I could have used 2.5 as
well, but I like 5/2.

00:02:48.280 --> 00:02:50.220
Our y-intercept we already
know is 0.

00:02:50.220 --> 00:02:51.700
There's no y-intercept here.

00:02:51.700 --> 00:02:54.050
You could do this as plus 0.

00:02:54.050 --> 00:02:55.990
So you start over here
at the origin.

00:02:55.990 --> 00:02:57.380
That's our y-intercept.

00:02:57.380 --> 00:03:00.280
And for every 2 that we go to
the right, we move up 5.

00:03:00.280 --> 00:03:02.240
So we move-- change in x is 2.

00:03:02.240 --> 00:03:05.120
One, two, three, four, five.

00:03:05.120 --> 00:03:06.180
Change in x is 2.

00:03:06.180 --> 00:03:09.160
One, two, three, four, five.

00:03:09.160 --> 00:03:11.550
If change in x is negative 2,
then change in y, it'll be

00:03:11.550 --> 00:03:12.630
negative 5.

00:03:12.630 --> 00:03:13.540
Negative 2.

00:03:13.540 --> 00:03:16.920
One, two, three, four, five.

00:03:16.920 --> 00:03:18.300
And so on and so forth.

00:03:18.300 --> 00:03:19.300
We'll eventually
get down here.

00:03:19.300 --> 00:03:23.755
So our line, our model if we
graph it, looks like this.

00:03:23.755 --> 00:03:26.800
I'm trying my best to go through
all of the points.

00:03:26.800 --> 00:03:28.360
Actually, I just said
I would only do it

00:03:28.360 --> 00:03:30.050
in the first quadrant.

00:03:30.050 --> 00:03:32.810
It really doesn't make sense in
this quadrant right here,

00:03:32.810 --> 00:03:34.900
because you can't have
negative minutes.

00:03:34.900 --> 00:03:37.370
Or you shouldn't have
negative minutes.

00:03:37.370 --> 00:03:40.240
So we should only be dealing
with it up here.

00:03:40.240 --> 00:03:43.560
Now they're asking how long
will it take to fill a

00:03:43.560 --> 00:03:45.540
30-gallon bathtub?

00:03:45.540 --> 00:03:48.540
Now, unfortunately, my graph
right here does not go all the

00:03:48.540 --> 00:03:49.510
way up to 30 gallons.

00:03:49.510 --> 00:03:52.800
If it did-- this is 10 gallons
right up here.

00:03:52.800 --> 00:03:55.390
That's 10 gallons if I went 3
times higher, I could just

00:03:55.390 --> 00:03:56.370
read the graph.

00:03:56.370 --> 00:03:59.780
But we could also solve for it
algebraically right here.

00:03:59.780 --> 00:04:01.150
How many minutes does it take?

00:04:01.150 --> 00:04:03.870
Well, let's just make our
gallons equal to 30.

00:04:03.870 --> 00:04:09.450
So you have 30 gallons is equal
to, I'll put it in the

00:04:09.450 --> 00:04:18.279
same color, 2.5 gallons per
minute times minutes.

00:04:21.510 --> 00:04:24.800
Now, all we have to do to solve
for minutes is divide

00:04:24.800 --> 00:04:28.400
both sides by 2.5 gallons
per minute.

00:04:28.400 --> 00:04:34.950
Divide both sides by 2.5
gallons per minute.

00:04:41.040 --> 00:04:43.080
I'm doing the units to show you
that the units actually

00:04:43.080 --> 00:04:43.970
all work out in the end.

00:04:43.970 --> 00:04:45.170
So this will cancel out.

00:04:45.170 --> 00:04:47.280
It will just become a 1.

00:04:47.280 --> 00:04:52.450
And so the left-hand side-- or
we could say m-- is going to

00:04:52.450 --> 00:04:57.100
be equal to 30 divided by 2.5.

00:04:57.100 --> 00:04:58.770
We have this gallons
in the numerator.

00:05:01.890 --> 00:05:03.580
I want to show you, you can deal
with the units just like

00:05:03.580 --> 00:05:06.180
you would deal with
actual numbers.

00:05:06.180 --> 00:05:08.570
And if I have gallons per minute
in the denominator, if

00:05:08.570 --> 00:05:10.730
I divide by this fraction,
that's the same thing is

00:05:10.730 --> 00:05:14.840
multiplying by its inverse.

00:05:14.840 --> 00:05:17.780
That's the same thing
as multiplying

00:05:17.780 --> 00:05:23.230
by minutes per gallon.

00:05:23.230 --> 00:05:24.020
Right?

00:05:24.020 --> 00:05:24.960
These units were in
the denominator.

00:05:24.960 --> 00:05:27.850
When I put them in the
numerator, I flip them.

00:05:27.850 --> 00:05:31.070
So gallons in the numerator,
gallons in the denominator.

00:05:31.070 --> 00:05:32.200
They cancel out.

00:05:32.200 --> 00:05:37.280
So I'm left with 30 divided
by 2.5 minutes.

00:05:37.280 --> 00:05:44.900
And what is 30 divided by 2.5?

00:05:44.900 --> 00:05:47.110
Is equal to 12.

00:05:47.110 --> 00:05:52.020
So it'll take us 12 minutes to
fill up a 30 gallon bathtub.

00:05:52.020 --> 00:05:53.270
We have our minutes
right there.

00:05:53.270 --> 00:05:54.880
12 minutes.

00:05:54.880 --> 00:05:57.850
Let's do one more.

00:05:57.850 --> 00:06:00.940
Amen is-- or maybe A-man, I
don't know the best way to

00:06:00.940 --> 00:06:03.750
pronounce that name-- is using
a hose-- let me scroll over a

00:06:03.750 --> 00:06:06.880
little bit-- is using a hose to
fill his new swimming pool

00:06:06.880 --> 00:06:08.070
for the first time.

00:06:08.070 --> 00:06:10.850
He starts the hose at 10 PM--
let me write this down.

00:06:10.850 --> 00:06:18.420
He starts at 10 PM-- so this is
the start time-- and leaves

00:06:18.420 --> 00:06:19.540
it running all night.

00:06:19.540 --> 00:06:22.150
At 6 AM, he measures the
depth and calculates

00:06:22.150 --> 00:06:24.520
the pool is 4/7 full.

00:06:24.520 --> 00:06:28.300
So when he starts at 10 PM--
so this is time and this is

00:06:28.300 --> 00:06:30.470
how full the pool is.

00:06:30.470 --> 00:06:32.680
So, obviously when he starts
it, the pool is empty.

00:06:32.680 --> 00:06:33.760
It's a new swimming pool.

00:06:33.760 --> 00:06:34.240
They tell us.

00:06:34.240 --> 00:06:37.530
So the pool is 0 full.

00:06:37.530 --> 00:06:39.300
It's 0 whatever.

00:06:39.300 --> 00:06:42.500
It has no water in
it whatsoever.

00:06:42.500 --> 00:06:49.790
Then at 6 AM, he measures the
depth and calculates the pool

00:06:49.790 --> 00:06:51.660
is 4/7 full.

00:06:51.660 --> 00:06:54.920
So here it is 4/7 full.

00:06:54.920 --> 00:06:58.160
At what time will his
new pool be full?

00:06:58.160 --> 00:06:59.530
So we want to know
when it's full.

00:06:59.530 --> 00:07:01.110
When it is 1/1 full?

00:07:01.110 --> 00:07:02.500
Where it's 7/7 full.

00:07:02.500 --> 00:07:04.420
At what time?

00:07:04.420 --> 00:07:07.290
So to do this, we have to set up
a similar model that we did

00:07:07.290 --> 00:07:08.500
the last time.

00:07:08.500 --> 00:07:13.770
We could say, the fullness of
the pool is equal to some

00:07:13.770 --> 00:07:17.170
constant times the amount of
time that's passed by.

00:07:21.320 --> 00:07:24.730
We know when time is equal to
0-- let me put this this way.

00:07:24.730 --> 00:07:25.910
This is time.

00:07:25.910 --> 00:07:26.490
Let me write it here.

00:07:26.490 --> 00:07:27.980
This is time.

00:07:27.980 --> 00:07:29.870
This is time, 0.

00:07:29.870 --> 00:07:31.920
What is this in hours?

00:07:31.920 --> 00:07:34.100
This is 8 hours later, right?

00:07:34.100 --> 00:07:35.640
This is time 8.

00:07:35.640 --> 00:07:37.150
We don't know what this is.

00:07:37.150 --> 00:07:38.660
This is time something else.

00:07:38.660 --> 00:07:44.320
So when time is 0 at 10 PM, 0
times k, we have 0 fullness.

00:07:44.320 --> 00:07:46.240
We are not full at all.

00:07:46.240 --> 00:07:53.180
When time is equal to 8, we have
k times 8. k is the rate

00:07:53.180 --> 00:07:55.210
at which we're filling
the pool.

00:07:55.210 --> 00:07:56.060
k times 8.

00:07:56.060 --> 00:07:58.600
We're at 4/7 fullness.

00:07:58.600 --> 00:08:01.490
So now we can actually
figure out what k is.

00:08:01.490 --> 00:08:04.480
We can figure out what our
proportionality constant is

00:08:04.480 --> 00:08:06.460
for our direct variation
model.

00:08:06.460 --> 00:08:10.790
Sounds very fancy, but all we're
saying is, look, this

00:08:10.790 --> 00:08:13.120
pool-filling business
can be modeled by an

00:08:13.120 --> 00:08:14.550
equation like this.

00:08:14.550 --> 00:08:17.200
The amount that we're full is
directly proportional to the

00:08:17.200 --> 00:08:19.790
amount of time that we
let the hose run.

00:08:19.790 --> 00:08:21.490
And this is the proportionality
constant.

00:08:21.490 --> 00:08:24.280
We don't know how quickly it
fills, but now we can figure

00:08:24.280 --> 00:08:25.800
out how quickly it fills.

00:08:25.800 --> 00:08:31.890
Because we know after 8
hours it is 4/7 full.

00:08:31.890 --> 00:08:35.169
So to solve for k, you divide
both sides by 8 hours.

00:08:35.169 --> 00:08:45.370
So we get k is equal to 4/7 full
divided by 8 hours, which

00:08:45.370 --> 00:08:53.150
is the same thing as 4/7 times
1/8 full per hour.

00:08:53.150 --> 00:08:56.140
So if we figure this
out, let's see.

00:08:56.140 --> 00:08:56.980
Divide by 4.

00:08:56.980 --> 00:08:58.360
Divide by 4.

00:08:58.360 --> 00:09:02.050
So we get 1/14 fullness.

00:09:02.050 --> 00:09:04.420
It's kind of a weird unit--
full per hour.

00:09:04.420 --> 00:09:08.690
Or you could say, we fill 1/14
of the pool per hour.

00:09:08.690 --> 00:09:10.920
So k is 1/14.

00:09:10.920 --> 00:09:15.240
So here our equation-- let
me write over here.

00:09:15.240 --> 00:09:23.300
The fullness of the pool is
equal to 1/14 times the time.

00:09:23.300 --> 00:09:25.130
So the question that we
have to answer is when

00:09:25.130 --> 00:09:28.240
does this equal 1?

00:09:28.240 --> 00:09:29.910
At what time?

00:09:29.910 --> 00:09:31.730
So let's set up the equation.

00:09:31.730 --> 00:09:32.720
So we have 1.

00:09:32.720 --> 00:09:34.035
That means we're completely
full.

00:09:34.035 --> 00:09:38.030
It is equal to 1/14
times time.

00:09:38.030 --> 00:09:44.520
If we multiply both sides of
this equation times 14, the

00:09:44.520 --> 00:09:46.420
1/14 and the 14 cancel out.

00:09:46.420 --> 00:09:50.950
And we're left with
t is equal to 14.

00:09:50.950 --> 00:09:54.240
So the pool will fill
after 14 hours.

00:09:54.240 --> 00:09:56.490
Remember everything we were
dealing with was hours.

00:09:56.490 --> 00:09:59.870
If we start at 10 PM-- that was
time 0-- what time are we

00:09:59.870 --> 00:10:01.860
at 14 hours?

00:10:01.860 --> 00:10:06.240
So 10 PM in one day-- if you
go to 10 AM the next day,

00:10:06.240 --> 00:10:08.280
that's 12 hours.

00:10:08.280 --> 00:10:09.010
That's how I think.

00:10:09.010 --> 00:10:14.600
We need to go 2 more hours
to get to 14 hours.

00:10:14.600 --> 00:10:18.670
That's noon the next day.

00:10:18.670 --> 00:10:22.490
At noon the day after he starts
filling is when the

00:10:22.490 --> 00:10:24.190
pool will be full.

00:10:24.190 --> 00:10:25.260
We can graph this.

00:10:25.260 --> 00:10:26.290
I had this graph paper here.

00:10:26.290 --> 00:10:28.820
Let's actually graph everything
I'm talking about.

00:10:28.820 --> 00:10:32.470
The equation-- I wrote it over
here-- fullness is equal to

00:10:32.470 --> 00:10:35.620
1/14 times t.

00:10:35.620 --> 00:10:37.860
Let's assume that each of
these notches is two.

00:10:37.860 --> 00:10:44.690
That is two, four, six, eight,
ten, twelve, fourteen.

00:10:44.690 --> 00:10:48.990
So this tells us that as
we run 14, we rise 1.

00:10:48.990 --> 00:10:54.360
So if x-change is 14, y-change
is positive 1.

00:10:54.360 --> 00:10:56.360
And I'll make these
units 1, 2.

00:10:56.360 --> 00:10:58.600
So the scale isn't
exactly perfect.

00:10:58.600 --> 00:11:01.370
I'm distorting the graph a
little bit, but that is 1.

00:11:01.370 --> 00:11:04.480
So the graph will look
something like that.

00:11:04.480 --> 00:11:05.340
Just like that.

00:11:05.340 --> 00:11:07.690
It has a slope of 1/14.

00:11:07.690 --> 00:11:10.210
Anyway, hopefully you
found that useful.


Slope-intercept equation from a graph examples

WEBVTT
Kind: captions
Language: en

00:00:00.720 --> 00:00:04.360
So you may or may not already
know that any linear equation

00:00:04.360 --> 00:00:10.300
can be written in the form
y is equal to mx plus b.

00:00:10.300 --> 00:00:12.630
Where m is the slope
of the line.

00:00:12.630 --> 00:00:14.590
The same slope that we've
been dealing with

00:00:14.590 --> 00:00:15.580
the last few videos.

00:00:15.580 --> 00:00:17.300
The rise over run of the line.

00:00:17.300 --> 00:00:19.090
Or the inclination
of the line.

00:00:19.090 --> 00:00:20.920
And b is the y-intercept.

00:00:23.890 --> 00:00:26.630
I think it's pretty easy
to verify that b is a

00:00:26.630 --> 00:00:27.310
y-intercept.

00:00:27.310 --> 00:00:30.350
The way you verify that is you
substitute x is equal to 0.

00:00:30.350 --> 00:00:35.590
If you get x is equal to 0--
remember x is equal to 0, that

00:00:35.590 --> 00:00:38.580
means that's where we're going
to intercept at the y-axis.

00:00:38.580 --> 00:00:42.630
If x is equal to 0, this
equation becomes y is equal to

00:00:42.630 --> 00:00:46.650
m times 0 plus b.

00:00:46.650 --> 00:00:48.430
m times 0 is just
going to be 0.

00:00:48.430 --> 00:00:49.520
I don't care what m is.

00:00:49.520 --> 00:00:53.620
So then y is going
to be equal to b.

00:00:53.620 --> 00:00:58.075
So the point 0, b is going
to be on that line.

00:01:00.680 --> 00:01:04.510
The line will intercept the
y-axis at the point

00:01:04.510 --> 00:01:05.540
y is equal to b.

00:01:05.540 --> 00:01:07.060
We'll see that with
actual numbers in

00:01:07.060 --> 00:01:08.420
the next few videos.

00:01:08.420 --> 00:01:12.080
Just to verify for you that m
is really the slope, let's

00:01:12.080 --> 00:01:13.310
just try some numbers out.

00:01:13.310 --> 00:01:16.840
We know the point 0,
b is on the line.

00:01:16.840 --> 00:01:18.465
What happens when
x is equal to 1?

00:01:21.700 --> 00:01:25.450
You get y is equal
to m times 1.

00:01:25.450 --> 00:01:28.210
Or it's equal to m plus b.

00:01:28.210 --> 00:01:34.200
So we also know that the
point 1, m plus b

00:01:34.200 --> 00:01:36.290
is also on the line.

00:01:36.290 --> 00:01:36.490
Right?

00:01:36.490 --> 00:01:37.900
This is just the y value.

00:01:37.900 --> 00:01:40.770
So what's the slope between
that point and that point?

00:01:40.770 --> 00:01:44.280
Let's take this as the end
point, so you have m plus b,

00:01:44.280 --> 00:01:52.030
our change in y, m plus b minus
b over our change in x,

00:01:52.030 --> 00:01:59.770
over 1 minus 0.

00:01:59.770 --> 00:02:02.740
This is our change in
y over change in x.

00:02:02.740 --> 00:02:03.660
We're using two points.

00:02:03.660 --> 00:02:04.480
That's our end point.

00:02:04.480 --> 00:02:06.120
That's our starting point.

00:02:06.120 --> 00:02:09.100
So if you simplify this,
b minus b is 0.

00:02:09.100 --> 00:02:10.840
1 minus 0 is 1.

00:02:10.840 --> 00:02:13.500
So you get m/1, or you
get it's equal to m.

00:02:13.500 --> 00:02:15.510
So hopefully you're satisfied
and hopefully I didn't confuse

00:02:15.510 --> 00:02:17.410
you by stating it in the
abstract with all of these

00:02:17.410 --> 00:02:18.410
variables here.

00:02:18.410 --> 00:02:21.140
But this is definitely going
to be the slope and this is

00:02:21.140 --> 00:02:23.580
definitely going to be
the y-intercept.

00:02:23.580 --> 00:02:27.360
Now given that, what I want to
do in this exercise is look at

00:02:27.360 --> 00:02:32.890
these graphs and then use the
already drawn graphs to figure

00:02:32.890 --> 00:02:33.860
out the equation.

00:02:33.860 --> 00:02:35.550
So we're going to look at these,
figure out the slopes,

00:02:35.550 --> 00:02:38.650
figure out the y-intercepts and
then know the equation.

00:02:38.650 --> 00:02:48.440
So let's do this line A first.
So what is A's slope?

00:02:48.440 --> 00:02:52.670
Let's start at some
arbitrary point.

00:02:52.670 --> 00:02:56.620
Let's start right over there.

00:02:56.620 --> 00:02:58.430
We want to get even numbers.

00:02:58.430 --> 00:03:03.070
If we run one, two, three.

00:03:03.070 --> 00:03:06.550
So if delta x is equal to 3.

00:03:06.550 --> 00:03:06.820
Right?

00:03:06.820 --> 00:03:07.550
One, two, three.

00:03:07.550 --> 00:03:09.640
Our delta y-- and I'm just doing
it because I want to hit

00:03:09.640 --> 00:03:15.430
an even number here-- our delta
y is equal to-- we go

00:03:15.430 --> 00:03:18.330
down by 2-- it's equal
to negative 2.

00:03:18.330 --> 00:03:22.790
So for A, change in
y for change in x.

00:03:22.790 --> 00:03:28.400
When our change in x is 3, our
change in y is negative 2.

00:03:28.400 --> 00:03:30.860
So our slope is negative 2/3.

00:03:30.860 --> 00:03:34.790
When we go over by 3, we're
going to go down by 2.

00:03:34.790 --> 00:03:37.920
Or if we go over by 1, we're
going to go down by 2/3.

00:03:37.920 --> 00:03:39.790
You can't exactly see it there,
but you definitely see

00:03:39.790 --> 00:03:41.800
it when you go over by 3.

00:03:41.800 --> 00:03:43.430
So that's our slope.

00:03:43.430 --> 00:03:45.380
We've essentially done
half of that problem.

00:03:45.380 --> 00:03:47.280
Now we have to figure
out the y-intercept.

00:03:47.280 --> 00:03:49.580
So that right there is our m.

00:03:49.580 --> 00:03:50.810
Now what is our b?

00:03:50.810 --> 00:03:52.340
Our y-intercept.

00:03:52.340 --> 00:03:56.120
Well where does this intersect
the y-axis?

00:03:56.120 --> 00:03:58.260
Well we already said
the slope is 2/3.

00:03:58.260 --> 00:04:01.250
So this is the point
y is equal to 2.

00:04:01.250 --> 00:04:05.610
When we go over by 1 to the
right, we would have

00:04:05.610 --> 00:04:07.880
gone down by 2/3.

00:04:07.880 --> 00:04:11.000
So this right here must
be the point 1 1/3.

00:04:11.000 --> 00:04:13.370
Or another way to say it,
we could say it's 4/3.

00:04:13.370 --> 00:04:15.920
That's the point y
is equal to 4/3.

00:04:15.920 --> 00:04:16.660
Right there.

00:04:16.660 --> 00:04:17.560
A little bit more than 1.

00:04:17.560 --> 00:04:19.269
About 1 1/3.

00:04:19.269 --> 00:04:22.029
So we could say b
is equal to 4/3.

00:04:22.029 --> 00:04:27.300
So we'll know that the equation
is y is equal to m,

00:04:27.300 --> 00:04:32.850
negative 2/3, x plus
b, plus 4/3.

00:04:32.850 --> 00:04:34.590
That's equation A.

00:04:34.590 --> 00:04:36.600
Let's do equation B.

00:04:36.600 --> 00:04:37.860
Hopefully we won't have
to deal with as

00:04:37.860 --> 00:04:39.510
many fractions here.

00:04:39.510 --> 00:04:40.360
Equation B.

00:04:40.360 --> 00:04:43.420
Let's figure out its slope
first. Let's start at some

00:04:43.420 --> 00:04:45.830
reasonable point.

00:04:45.830 --> 00:04:48.510
We could start at that point.

00:04:48.510 --> 00:04:49.650
Let me do it right here.

00:04:49.650 --> 00:04:50.380
B.

00:04:50.380 --> 00:04:52.030
Equation B.

00:04:52.030 --> 00:04:56.100
When our delta x is equal
to-- let me write it

00:04:56.100 --> 00:04:57.960
this way, delta x.

00:04:57.960 --> 00:05:00.040
So our delta x could be 1.

00:05:00.040 --> 00:05:02.640
When we move over 1 to the
right, what happens

00:05:02.640 --> 00:05:03.480
to our delta y?

00:05:03.480 --> 00:05:06.270
We go up by 3.

00:05:06.270 --> 00:05:07.710
delta x.

00:05:07.710 --> 00:05:08.670
delta y.

00:05:08.670 --> 00:05:11.140
Our change in y is 3.

00:05:11.140 --> 00:05:14.710
So delta y over delta x, When
we go to the right, our

00:05:14.710 --> 00:05:17.940
change in x is 1.

00:05:17.940 --> 00:05:20.830
Our change in y is positive 3.

00:05:20.830 --> 00:05:23.760
So our slope is equal to 3.

00:05:23.760 --> 00:05:25.410
What is our y-intercept?

00:05:25.410 --> 00:05:29.550
Well, when x is equal to
0, y is equal to 1.

00:05:29.550 --> 00:05:32.160
So b is equal to 1.

00:05:32.160 --> 00:05:33.210
So this was a lot easier.

00:05:33.210 --> 00:05:39.430
Here the equation is y is
equal to 3x plus 1.

00:05:39.430 --> 00:05:41.280
Let's do that last line there.

00:05:41.280 --> 00:05:49.100
Line C Let's do the y-intercept
first. You see

00:05:49.100 --> 00:05:51.820
immediately the y-intercept--
when x is equal to 0, y is

00:05:51.820 --> 00:05:53.040
negative 2.

00:05:53.040 --> 00:05:55.380
So b is equal to negative 2.

00:05:55.380 --> 00:05:57.300
And then what is the slope?

00:05:57.300 --> 00:06:03.220
m is equal to change in
y over change in x.

00:06:03.220 --> 00:06:04.880
Let's start at that
y-intercept.

00:06:04.880 --> 00:06:09.630
If we go over to the right
by one, two, three, four.

00:06:09.630 --> 00:06:12.570
So our change in x
is equal to 4.

00:06:12.570 --> 00:06:14.380
What is our change in y?

00:06:14.380 --> 00:06:16.945
Our change in y is positive 2.

00:06:19.630 --> 00:06:23.780
So change in y is 2 when
change in x is 4.

00:06:23.780 --> 00:06:27.390
So the slope is equal
to 1/2, 2/4.

00:06:27.390 --> 00:06:31.870
So the equation here is y is
equal to 1/2 x, that's our

00:06:31.870 --> 00:06:34.750
slope, minus 2.

00:06:34.750 --> 00:06:36.560
And we're done.

00:06:36.560 --> 00:06:38.480
Now let's go the other way.

00:06:38.480 --> 00:06:41.850
Let's look at some equations of
lines knowing that this is

00:06:41.850 --> 00:06:44.900
the slope and this is the
y-intercept-- that's the m,

00:06:44.900 --> 00:06:47.340
that's the b-- and actually
graph them.

00:06:47.340 --> 00:06:48.320
Let's do this first line.

00:06:48.320 --> 00:06:50.030
I already started circling
it in orange.

00:06:50.030 --> 00:06:51.350
The y-intercept is 5.

00:06:51.350 --> 00:06:53.800
When x is equal to 0,
y is equal to 5.

00:06:53.800 --> 00:06:55.240
You can verify that
on the equation.

00:06:55.240 --> 00:06:58.920
So when x is equal to 0, y is
equal to one, two, three,

00:06:58.920 --> 00:07:00.590
four, five.

00:07:00.590 --> 00:07:03.890
That's the y-intercept
and the slope is 2.

00:07:03.890 --> 00:07:08.330
That means when I move 1 in the
x-direction, I move up 2

00:07:08.330 --> 00:07:09.770
in the y-direction.

00:07:09.770 --> 00:07:12.490
If I move 1 in the x-direction,
I move up 2 in

00:07:12.490 --> 00:07:12.990
the y-direction.

00:07:12.990 --> 00:07:16.050
If I move 1 in the x-direction,
I move up 2 in

00:07:16.050 --> 00:07:17.320
the y-direction.

00:07:17.320 --> 00:07:20.550
If I move back 1 in the
x-direction, I move down 2 in

00:07:20.550 --> 00:07:20.830
the y-direction.

00:07:20.830 --> 00:07:24.160
If I move back 1 in the
x-direction, I move down 2 in

00:07:24.160 --> 00:07:24.750
the y-direction.

00:07:24.750 --> 00:07:25.970
I keep doing that.

00:07:25.970 --> 00:07:29.130
So this line is going to look--
I can't draw lines too

00:07:29.130 --> 00:07:31.060
neatly, but this is going
to be my best shot.

00:07:31.060 --> 00:07:33.240
It's going to look something
like that.

00:07:33.240 --> 00:07:36.100
It'll just keep going on,
on and on and on.

00:07:36.100 --> 00:07:38.140
So that's our first line.

00:07:38.140 --> 00:07:40.720
I can just keep going
down like that.

00:07:40.720 --> 00:07:42.420
Let's do this second line.

00:07:42.420 --> 00:07:47.200
y is equal to negative
0.2x plus 7.

00:07:47.200 --> 00:07:53.910
Let me write that. y is equal
to negative 0.2x plus 7.

00:07:53.910 --> 00:07:55.450
It's always easier to
think in fractions.

00:07:55.450 --> 00:07:57.420
So 0.2 is the same
thing as 1/5.

00:07:57.420 --> 00:08:03.500
We could write y is equal to
negative 1/5 x plus 7.

00:08:03.500 --> 00:08:05.000
We know it's y-intercept at 7.

00:08:05.000 --> 00:08:08.270
So it's one, two, three,
four, five, six.

00:08:08.270 --> 00:08:11.100
That's our y-intercept
when x is equal to 0.

00:08:11.100 --> 00:08:15.240
This tells us that for every
5 we move to the right,

00:08:15.240 --> 00:08:18.120
we move down 1.

00:08:18.120 --> 00:08:21.330
We can view this as
negative 1/5.

00:08:21.330 --> 00:08:25.590
The delta y over delta x is
equal to negative 1/5.

00:08:25.590 --> 00:08:28.430
For every 5 we move to the
right, we move down 1.

00:08:28.430 --> 00:08:29.370
So every 5.

00:08:29.370 --> 00:08:31.550
One, two, three, four, five.

00:08:31.550 --> 00:08:32.650
We moved 5 to the right.

00:08:32.650 --> 00:08:35.169
That means we must
move down 1.

00:08:35.169 --> 00:08:36.309
We move 5 to the right.

00:08:36.309 --> 00:08:38.690
One, two, three, four, five.

00:08:38.690 --> 00:08:40.549
We must move down 1.

00:08:40.549 --> 00:08:44.740
If you go backwards, if you move
5 backwards-- instead of

00:08:44.740 --> 00:08:47.260
this, if you view this
as 1 over negative 5.

00:08:47.260 --> 00:08:49.450
These are obviously equivalent
numbers.

00:08:49.450 --> 00:08:52.060
If you go back 5-- that's
negative 5.

00:08:52.060 --> 00:08:54.500
One, two, three, four, five.

00:08:54.500 --> 00:08:56.805
Then you move up 1.

00:08:56.805 --> 00:09:00.050
If you go back 5-- one,
two, three, four,

00:09:00.050 --> 00:09:01.830
five-- you move up 1.

00:09:01.830 --> 00:09:04.110
So the line is going
to look like this.

00:09:04.110 --> 00:09:07.000
I have to just connect
the dots.

00:09:07.000 --> 00:09:08.020
I think you get the idea.

00:09:08.020 --> 00:09:09.170
I just have to connect
those dots.

00:09:09.170 --> 00:09:11.170
I could've drawn it a little
bit straighter.

00:09:11.170 --> 00:09:13.840
Now let's do this one, y
is equal to negative x.

00:09:16.480 --> 00:09:17.800
Where's the b term?

00:09:17.800 --> 00:09:18.940
I don't see any b term.

00:09:18.940 --> 00:09:22.490
You remember we're saying
y is equal to mx plus b.

00:09:22.490 --> 00:09:23.670
Where is the b?

00:09:23.670 --> 00:09:24.780
Well, the b is 0.

00:09:24.780 --> 00:09:26.770
You could view this as plus 0.

00:09:26.770 --> 00:09:28.290
Here is b is 0.

00:09:28.290 --> 00:09:31.710
When x is 0, y is 0.

00:09:31.710 --> 00:09:35.080
That's our y-intercept, right
there at the origin.

00:09:35.080 --> 00:09:37.490
And then the slope-- once again
you see a negative sign.

00:09:37.490 --> 00:09:40.710
You could view that as
negative 1x plus 0.

00:09:40.710 --> 00:09:42.200
So slope is negative 1.

00:09:42.200 --> 00:09:45.570
When you move to the right by
1, when change in x is 1,

00:09:45.570 --> 00:09:47.690
change in y is negative 1.

00:09:47.690 --> 00:09:52.320
When you move up by 1 in x,
you go down by 1 in y.

00:09:52.320 --> 00:09:55.580
Or if you go down by 1 in x,
you're going to go up by 1 in

00:09:55.580 --> 00:09:57.660
y. x and y are going to
have opposite signs.

00:09:57.660 --> 00:09:58.850
They go in opposite
directions.

00:09:58.850 --> 00:10:03.955
So the line is going
to look like that.

00:10:03.955 --> 00:10:06.550
You could almost imagine it's
splitting the second and

00:10:06.550 --> 00:10:08.240
fourth quadrants.

00:10:08.240 --> 00:10:09.980
Now I'll do one more.

00:10:09.980 --> 00:10:12.160
Let's do this last
one right here.

00:10:12.160 --> 00:10:16.830
y is equal to 3.75.

00:10:16.830 --> 00:10:19.610
Now you're saying, gee,
we're looking for y is

00:10:19.610 --> 00:10:21.630
equal to mx plus b.

00:10:21.630 --> 00:10:22.940
Where is this x term?

00:10:22.940 --> 00:10:24.650
It's completely gone.

00:10:24.650 --> 00:10:27.980
Well the reality here is, this
could be rewritten as y is

00:10:27.980 --> 00:10:33.240
equal to 0x plus 3.75.

00:10:33.240 --> 00:10:34.120
Now it makes sense.

00:10:34.120 --> 00:10:35.500
The slope is 0.

00:10:35.500 --> 00:10:39.020
No matter how much we change
our x, y does not change.

00:10:39.020 --> 00:10:42.930
Delta y over delta
x is equal to 0.

00:10:42.930 --> 00:10:45.770
I don't care how much
you change your x.

00:10:45.770 --> 00:10:48.220
Our y-intercept is 3.75.

00:10:48.220 --> 00:10:51.870
So 1, 2, 3.75 is right
around there.

00:10:51.870 --> 00:10:52.710
You want to get close.

00:10:52.710 --> 00:10:54.990
3 3/4.

00:10:54.990 --> 00:10:57.620
As I change x, y will
not change. y is

00:10:57.620 --> 00:11:00.570
always going to be 3.75.

00:11:00.570 --> 00:11:03.200
It's just going to be
a horizontal line at

00:11:03.200 --> 00:11:07.450
y is equal to 3.75.

00:11:07.450 --> 00:11:10.600
Anyway, hopefully you
found this useful.


Slope and rate of change

WEBVTT
Kind: captions
Language: en

00:00:00.410 --> 00:00:05.430
In this video I'm going to do
a bunch of example slope

00:00:05.430 --> 00:00:09.450
problems. Just as a bit of
review, slope is just a way of

00:00:09.450 --> 00:00:11.920
measuring the inclination
of a line.

00:00:11.920 --> 00:00:13.870
And the definition-- we're going
to hopefully get a good

00:00:13.870 --> 00:00:16.800
working knowledge of it in this
video-- the definition of

00:00:16.800 --> 00:00:21.720
it is a change in y divided
by change in x.

00:00:21.720 --> 00:00:24.960
This may or may not make some
sense to you right now, but as

00:00:24.960 --> 00:00:27.830
we do more and more examples,
I think it'll make a good

00:00:27.830 --> 00:00:28.730
amount of sense.

00:00:28.730 --> 00:00:30.900
Let's do this first
line right here.

00:00:30.900 --> 00:00:31.660
Line a.

00:00:31.660 --> 00:00:34.460
Let's figure out its slope.

00:00:34.460 --> 00:00:37.490
They've actually drawn two
points here that we can use as

00:00:37.490 --> 00:00:39.130
the reference points.

00:00:39.130 --> 00:00:40.140
So first of all, let's
look at the

00:00:40.140 --> 00:00:41.770
coordinates of those points.

00:00:41.770 --> 00:00:44.140
So you have this point
right here.

00:00:44.140 --> 00:00:45.040
What's its coordinates?

00:00:45.040 --> 00:00:48.480
Its x-coordinate is 3.

00:00:48.480 --> 00:00:51.070
Its y-coordinate is 6.

00:00:51.070 --> 00:00:54.940
And then down here, this
point's x-coordinate is

00:00:54.940 --> 00:01:00.780
negative 1 and its y-coordinate
is negative 6.

00:01:00.780 --> 00:01:02.920
So there's a couple of ways
we can think about slope.

00:01:02.920 --> 00:01:05.330
One is, we could look at it
straight up using the formula.

00:01:05.330 --> 00:01:10.030
We could say change in y-- so
slope is change in y over

00:01:10.030 --> 00:01:11.370
change in x.

00:01:11.370 --> 00:01:12.850
We can figure it out
numerically.

00:01:12.850 --> 00:01:15.200
I'll in a second draw
it graphically.

00:01:15.200 --> 00:01:17.040
So what's our change in y?

00:01:17.040 --> 00:01:20.330
Our change in y is literally
how much did our y values

00:01:20.330 --> 00:01:23.760
change going from this
point to that point?

00:01:23.760 --> 00:01:25.400
So how much did our
y values change?

00:01:25.400 --> 00:01:29.570
Our y went from here, y is at
negative 6 and it went all the

00:01:29.570 --> 00:01:34.060
way up to positive 6.

00:01:34.060 --> 00:01:36.150
So what's this distance
right here?

00:01:36.150 --> 00:01:38.930
It's going to be your
end point y value.

00:01:38.930 --> 00:01:44.040
It's going to be 6 minus your
starting point y value.

00:01:44.040 --> 00:01:50.140
Minus negative 6 or 6 plus
6, which is equal to 12.

00:01:50.140 --> 00:01:50.960
You could just count this.

00:01:50.960 --> 00:01:56.000
You say one, two, three, four,
five, six, seven, eight, nine,

00:01:56.000 --> 00:01:57.840
ten, eleven, twelve.

00:01:57.840 --> 00:02:02.680
So when we changed our y value
by 12, we had to change our x

00:02:02.680 --> 00:02:06.460
value by-- what was our
change it x over the

00:02:06.460 --> 00:02:07.800
same change in y?

00:02:07.800 --> 00:02:10.570
Well we went from x is equal
to negative 1 to

00:02:10.570 --> 00:02:14.190
x is equal to 3.

00:02:14.190 --> 00:02:16.930
Right? x went from
negative 1 to 3.

00:02:16.930 --> 00:02:22.110
So we do the end point, which is
3 minus the starting point,

00:02:22.110 --> 00:02:24.540
which is negative 1, which
is equal to 4.

00:02:24.540 --> 00:02:30.270
So our change in y over change
in x is equal to 12/4 or if we

00:02:30.270 --> 00:02:32.660
want to write this in simplest
form, this is the

00:02:32.660 --> 00:02:35.030
same thing as 3.

00:02:35.030 --> 00:02:38.690
Now the interpretation of this
means that for every 1 we move

00:02:38.690 --> 00:02:40.520
over-- we could view this,
let me write it this way.

00:02:40.520 --> 00:02:44.190
Change in y over change in x
is equal to-- we could say

00:02:44.190 --> 00:02:46.960
it's 3 or we could
say it's 3/1.

00:02:46.960 --> 00:02:49.590
Which tells us that for every
1 we move in the positive

00:02:49.590 --> 00:02:52.650
x-direction, we're going to move
up 3 because this is a

00:02:52.650 --> 00:02:54.540
positive 3 in the y-direction.

00:02:54.540 --> 00:02:55.480
You can see that.

00:02:55.480 --> 00:02:59.320
When we moved 1 in the x,
we moved up 3 in the y.

00:02:59.320 --> 00:03:02.880
When we moved 1 in the x,
we moved up 3 in the y.

00:03:02.880 --> 00:03:06.500
If you move 2 in the
x-direction, you're going to

00:03:06.500 --> 00:03:08.250
move 6 in the y.

00:03:08.250 --> 00:03:11.210
6/2 is the same thing as 3.

00:03:11.210 --> 00:03:17.280
So this 3 tells us how quickly
do we go up as we increase x.

00:03:17.280 --> 00:03:21.780
Let's do the same thing for the
second line on this graph.

00:03:21.780 --> 00:03:23.470
Graph b.

00:03:23.470 --> 00:03:24.060
Same idea.

00:03:24.060 --> 00:03:26.300
I'm going to use the points
that they gave us.

00:03:26.300 --> 00:03:28.730
But really you could use any
points on that line.

00:03:28.730 --> 00:03:31.020
So let's see, we have one
point here, which is

00:03:31.020 --> 00:03:34.550
the point 0, 1.

00:03:34.550 --> 00:03:36.540
You have 0, 1.

00:03:36.540 --> 00:03:38.780
And then the starting point-- we
could call this the finish

00:03:38.780 --> 00:03:41.460
point-- the starting point right
here, we could view it

00:03:41.460 --> 00:03:47.830
as x is negative 6 and
y is negative 2.

00:03:47.830 --> 00:03:48.990
So same idea.

00:03:48.990 --> 00:03:53.230
What is the change in y given
some change in x?

00:03:53.230 --> 00:03:56.440
So let's do the change in
x first. So what is

00:03:56.440 --> 00:03:58.060
our change in x?

00:03:58.060 --> 00:04:02.930
So in this situation, what is
our change in x? delta x.

00:04:02.930 --> 00:04:03.850
We could even count it.

00:04:03.850 --> 00:04:06.910
It's one, two, three,
four, five, six.

00:04:06.910 --> 00:04:08.110
It's going to be 6.

00:04:08.110 --> 00:04:10.760
But if you didn't have a graph
to count from, you could

00:04:10.760 --> 00:04:16.370
literally take your finishing
x-position, so it's 0, and

00:04:16.370 --> 00:04:19.029
subtract from that your
starting x-position.

00:04:19.029 --> 00:04:22.390
0 minus negative 6.

00:04:22.390 --> 00:04:26.360
So when your change in x is
equal to-- so this will be 6--

00:04:26.360 --> 00:04:29.180
what is our change in y?

00:04:29.180 --> 00:04:31.890
Remember we're taking this as
our finishing position.

00:04:36.200 --> 00:04:38.520
This is our starting position.

00:04:38.520 --> 00:04:40.670
So we took 0 minus negative 6.

00:04:40.670 --> 00:04:43.635
So then on the y, we have to
do 1 minus negative 2.

00:04:47.680 --> 00:04:49.180
What's 1 minus negative 2?

00:04:49.180 --> 00:04:51.260
That's the same thing
as 1 plus 2.

00:04:51.260 --> 00:04:52.410
That is equal to 3.

00:04:52.410 --> 00:04:56.450
So it is 3/6 or 1/2.

00:04:56.450 --> 00:05:02.080
So notice, when we moved in the
x-direction by 6, we moved

00:05:02.080 --> 00:05:05.340
in the y-direction
by positive 3.

00:05:05.340 --> 00:05:09.170
So our change in y was 3 when
our change in x was 6.

00:05:09.170 --> 00:05:11.430
Now, one of the things that
confuses a lot of people is

00:05:11.430 --> 00:05:16.110
how do I know what order to--
how did I know to do the 0

00:05:16.110 --> 00:05:18.930
first and the negative 6 second
and then the 1 first

00:05:18.930 --> 00:05:19.920
and then the negative
2 second.

00:05:19.920 --> 00:05:22.210
And the answer is you could've
done it in either order as

00:05:22.210 --> 00:05:23.540
long as you keep
them straight.

00:05:23.540 --> 00:05:27.300
So you could have also
have done change in y

00:05:27.300 --> 00:05:28.830
over change in x.

00:05:28.830 --> 00:05:34.280
We could have said, it's equal
to negative 2 minus 1.

00:05:34.280 --> 00:05:37.690
So we're using this coordinate
first. Negative 2 minus 1 for

00:05:37.690 --> 00:05:43.490
the y over negative 6 minus 0.

00:05:43.490 --> 00:05:45.960
Notice this is a negative
of that.

00:05:45.960 --> 00:05:47.650
That is the negative of that.

00:05:47.650 --> 00:05:49.545
But since we have a negative
over negative, they're going

00:05:49.545 --> 00:05:50.260
to cancel out.

00:05:50.260 --> 00:05:54.810
So this is going to be equal to
negative 3 over negative 6.

00:05:54.810 --> 00:05:56.020
The negatives cancel out.

00:05:56.020 --> 00:05:58.250
This is also equal to 1/2.

00:05:58.250 --> 00:06:05.250
So the important thing is if
you use this y-coordinate

00:06:05.250 --> 00:06:09.280
first, then you have
to use this

00:06:09.280 --> 00:06:11.350
x-coordinate first as well.

00:06:11.350 --> 00:06:15.640
If you use this y-coordinate
first, as we did here, then

00:06:15.640 --> 00:06:17.160
you have to use this
x-coordinate

00:06:17.160 --> 00:06:18.130
first, as you did there.

00:06:18.130 --> 00:06:19.850
You just have to make sure
that your change in x and

00:06:19.850 --> 00:06:22.750
change in y are-- you're
using the same final

00:06:22.750 --> 00:06:23.950
and starting points.

00:06:23.950 --> 00:06:30.190
Just to interpret this, this is
saying that for every minus

00:06:30.190 --> 00:06:31.820
6 we go in x.

00:06:31.820 --> 00:06:36.480
So if we go minus 6 in x, so
that's going backwards, we're

00:06:36.480 --> 00:06:40.210
going to go minus 3 in y.

00:06:40.210 --> 00:06:41.960
But they're essentially
saying the same thing.

00:06:41.960 --> 00:06:44.170
The slope of this line is 1/2.

00:06:44.170 --> 00:06:50.150
Which tells us for every 2 we
travel in x, we go up 1 in y.

00:06:50.150 --> 00:06:53.990
Or if we go back 2 in x,
we go down 1 in y.

00:06:53.990 --> 00:06:56.070
That's what 1/2 slope
tells us.

00:06:56.070 --> 00:07:00.560
Notice, the line with the 1/2
slope, it is less steep than

00:07:00.560 --> 00:07:03.840
the line with a slope of 3.

00:07:03.840 --> 00:07:06.490
Let's do a couple
more of these.

00:07:06.490 --> 00:07:11.200
Let's do line c right here.

00:07:11.200 --> 00:07:12.670
I'll do it in pink.

00:07:12.670 --> 00:07:15.250
Let's say that the starting
point-- I'm just picking this

00:07:15.250 --> 00:07:16.120
arbitrarily.

00:07:16.120 --> 00:07:18.840
Well, I'm using these points
that they've drawn here.

00:07:18.840 --> 00:07:23.010
The starting point is at the
coordinate negative 1, 6 and

00:07:23.010 --> 00:07:29.050
that my finishing point is at
the point 5, negative 6.

00:07:32.940 --> 00:07:35.980
Our slope is going to be-- let
me write this-- slope is going

00:07:35.980 --> 00:07:40.480
to be equal to change in
x-- sorry, change in y.

00:07:40.480 --> 00:07:41.460
I'll never forget that.

00:07:41.460 --> 00:07:43.500
Change in y over change in x.

00:07:43.500 --> 00:07:46.170
Sometimes it's said
rise over run.

00:07:46.170 --> 00:07:48.610
Run is how much you're moving
in the horizontal direction.

00:07:48.610 --> 00:07:51.360
Rise is how much you're moving
in the vertical direction.

00:07:51.360 --> 00:07:55.840
Then we could say our change in
y is our finishing y-point

00:07:55.840 --> 00:07:59.560
minus our starting y-point.

00:07:59.560 --> 00:08:01.990
This is our finishing y-point.

00:08:01.990 --> 00:08:05.990
That's our starting y-point,
over our finishing x-point

00:08:05.990 --> 00:08:08.800
minus our starting x-point.

00:08:08.800 --> 00:08:11.440
If that confuses you, all I'm
saying is, it's going to be

00:08:11.440 --> 00:08:16.570
equal to our finishing y-point
is negative 6 minus our

00:08:16.570 --> 00:08:21.990
starting y-point, which is 6,
over our finishing x-point,

00:08:21.990 --> 00:08:27.380
which is 5, minus our starting
x-point, which is negative 1.

00:08:27.380 --> 00:08:31.780
So this is equal to negative
6 minus 6 is negative 12.

00:08:31.780 --> 00:08:34.280
5 minus negative 1.

00:08:34.280 --> 00:08:36.080
That is 6.

00:08:36.080 --> 00:08:37.840
So negative 12/6.

00:08:37.840 --> 00:08:41.240
That's the same thing
as negative 2.

00:08:41.240 --> 00:08:45.230
Notice we have a negative
slope here.

00:08:45.230 --> 00:08:49.670
That's because every time we
increase x by 1, we go down in

00:08:49.670 --> 00:08:50.740
the y-direction.

00:08:50.740 --> 00:08:52.730
So this is a downward
sloping line.

00:08:52.730 --> 00:08:55.170
It's going from the top left
to the bottom right.

00:08:55.170 --> 00:08:58.610
As x increases, the
y decreases.

00:08:58.610 --> 00:09:00.130
And that's why we got
a negative slope.

00:09:00.130 --> 00:09:03.010
This line over here should
have a positive slope.

00:09:03.010 --> 00:09:04.860
Let's verify it.

00:09:04.860 --> 00:09:07.360
So I'll use the same
points that they

00:09:07.360 --> 00:09:09.400
use right over there.

00:09:09.400 --> 00:09:11.760
So this is line d.

00:09:11.760 --> 00:09:17.760
Slope is equal to
rise over run.

00:09:17.760 --> 00:09:21.140
How much do we rise when we go
from that point to that point?

00:09:21.140 --> 00:09:21.570
Let's see.

00:09:21.570 --> 00:09:22.270
We could do it this way.

00:09:22.270 --> 00:09:25.500
We are rising-- I could
just count it out.

00:09:25.500 --> 00:09:30.110
We are rising one, two, three,
four, five, six.

00:09:30.110 --> 00:09:32.460
We are rising 6.

00:09:32.460 --> 00:09:33.780
How much are we running?

00:09:33.780 --> 00:09:36.150
We are running-- I'll do it
in a different color.

00:09:36.150 --> 00:09:41.580
We're running one, two, three,
four, five, six.

00:09:41.580 --> 00:09:43.640
We're running 6.

00:09:43.640 --> 00:09:46.300
So our slope is 6/6,
which is 1.

00:09:46.300 --> 00:09:49.510
Which tells us that every time
we move 1 in the x-direction--

00:09:49.510 --> 00:09:52.390
positive 1 in the x-direction--
we go positive 1

00:09:52.390 --> 00:09:55.390
in the y-direction.

00:09:55.390 --> 00:09:59.340
For every x, if we go negative
2 in the x-direction, we're

00:09:59.340 --> 00:10:02.340
going to go negative 2
in the y-direction.

00:10:02.340 --> 00:10:04.820
So whatever we do in x, we're
going to do the same thing in

00:10:04.820 --> 00:10:05.710
y in this slope.

00:10:05.710 --> 00:10:06.740
Notice, that was pretty easy.

00:10:06.740 --> 00:10:09.200
If we wanted to do it
mathematically, we could

00:10:09.200 --> 00:10:11.260
figure out this coordinate
right there.

00:10:11.260 --> 00:10:13.570
That we could view as our
starting position.

00:10:13.570 --> 00:10:17.630
Our starting position is
negative 2, negative 4.

00:10:20.350 --> 00:10:26.210
Our finishing position
is 4, 2.

00:10:29.720 --> 00:10:34.850
So our slope, change in
y over change in x.

00:10:34.850 --> 00:10:42.000
I'll take this point 2 minus
negative 4 over 4 minus

00:10:42.000 --> 00:10:43.250
negative 2.

00:10:45.450 --> 00:10:48.970
2 minus negative 4 is 6.

00:10:48.970 --> 00:10:52.000
Remember that was just this
distance right there.

00:10:52.000 --> 00:10:55.140
Then 4 minus negative
2, that's also 6.

00:10:55.140 --> 00:10:57.300
That's that distance
right there.

00:10:57.300 --> 00:11:00.600
We get a slope of 1.

00:11:00.600 --> 00:11:01.380
Let's do another one.

00:11:01.380 --> 00:11:02.210
Let's do another couple.

00:11:02.210 --> 00:11:03.940
These are interesting.

00:11:03.940 --> 00:11:08.200
Let's do the line
e right here.

00:11:08.200 --> 00:11:11.800
Change in y over change in x.

00:11:11.800 --> 00:11:14.420
So our change in y, when we
go from this point to this

00:11:14.420 --> 00:11:16.160
point-- I'll just
count it out.

00:11:16.160 --> 00:11:19.210
It's one, two, three, four,
five, six, seven, eight.

00:11:19.210 --> 00:11:20.530
It's 8.

00:11:20.530 --> 00:11:25.510
Or you could even take this
y-coordinate 2 minus negative

00:11:25.510 --> 00:11:28.460
6 will give you that
distance, 8.

00:11:28.460 --> 00:11:30.020
What's the change in y?

00:11:30.020 --> 00:11:32.640
Well the y-value here is-- oh
sorry what's the change in x?

00:11:32.640 --> 00:11:34.260
The x-value here is 4.

00:11:34.260 --> 00:11:35.390
The x-value there is 4.

00:11:35.390 --> 00:11:36.630
X does not change.

00:11:36.630 --> 00:11:39.280
So it's 8/0.

00:11:39.280 --> 00:11:40.660
Well, we don't know.

00:11:40.660 --> 00:11:42.320
8/0 is undefined.

00:11:42.320 --> 00:11:44.630
So in this situation the
slope is undefined.

00:11:44.630 --> 00:11:46.830
When you have a vertical
line, you say

00:11:46.830 --> 00:11:49.235
your slope is undefined.

00:11:52.410 --> 00:11:54.190
Because you're dividing by 0.

00:11:54.190 --> 00:11:56.530
But that tells you that you're
dealing probably with a

00:11:56.530 --> 00:11:58.320
vertical line.

00:11:58.320 --> 00:11:59.720
Now finally let's just
do this one.

00:11:59.720 --> 00:12:02.670
This seems like a pretty
straight up vanilla slope

00:12:02.670 --> 00:12:04.320
problem right there.

00:12:04.320 --> 00:12:08.050
You have that point right
there, which is

00:12:08.050 --> 00:12:10.090
the point 3, 1.

00:12:10.090 --> 00:12:11.090
So this is line f.

00:12:11.090 --> 00:12:14.210
You have the point 3, 1.

00:12:14.210 --> 00:12:19.750
Then over here you have the
point negative 6, negative 2.

00:12:19.750 --> 00:12:24.440
So our slope would be equal
to change in y.

00:12:24.440 --> 00:12:28.960
I'll take this as our ending
point, just so you can go in

00:12:28.960 --> 00:12:30.130
different directions.

00:12:30.130 --> 00:12:31.770
So our change in y-- now
we're going to go

00:12:31.770 --> 00:12:33.820
down in that direction.

00:12:33.820 --> 00:12:36.540
So it's negative 2 minus 1.

00:12:36.540 --> 00:12:38.810
That's what this distance
is right here.

00:12:38.810 --> 00:12:42.240
Negative 2 minus 1, which
is equal to negative 3.

00:12:42.240 --> 00:12:44.340
Notice we went down 3.

00:12:44.340 --> 00:12:47.340
And then what is going to
be our change in x?

00:12:47.340 --> 00:12:50.480
Well, we're going to go
back that amount.

00:12:50.480 --> 00:12:51.880
What is that amount?

00:12:51.880 --> 00:12:56.200
Well, that is going to be
negative 6, that's our end

00:12:56.200 --> 00:12:59.920
point, minus 3.

00:12:59.920 --> 00:13:04.700
That gives us that distance,
which is negative 9.

00:13:04.700 --> 00:13:09.140
For every time we go back 9,
we're going to go down 3.

00:13:12.680 --> 00:13:16.065
Which is the same thing as if
we go forward 9, we're going

00:13:16.065 --> 00:13:17.370
to go up 3.

00:13:17.370 --> 00:13:18.390
All equivalent.

00:13:18.390 --> 00:13:21.830
And we see these cancel out and
you get a slope of 1/3.

00:13:21.830 --> 00:13:22.810
Positive 1/3.

00:13:22.810 --> 00:13:25.970
It's an upward sloping line.

00:13:25.970 --> 00:13:33.110
Every time we run
3, we rise 1.

00:13:37.740 --> 00:13:41.810
Anyway, hopefully that was a
good review of slope for you.


Graphs of linear equations

WEBVTT
Kind: captions
Language: en

00:00:00.390 --> 00:00:05.080
Let's do a couple of problems
graphing linear equations.

00:00:05.080 --> 00:00:07.280
They are a bunch of ways to
graph linear equations.

00:00:07.280 --> 00:00:09.560
What we'll do in this video
is the most basic way.

00:00:09.560 --> 00:00:11.790
Where we will just plot a
bunch of values and then

00:00:11.790 --> 00:00:12.750
connect the dots.

00:00:12.750 --> 00:00:13.830
I think you'll see
what I'm saying.

00:00:13.830 --> 00:00:16.460
So here I have an equation,
a linear equation.

00:00:16.460 --> 00:00:18.990
I'll rewrite it just in case
that was too small. y is equal

00:00:18.990 --> 00:00:22.330
to 2x plus 7.

00:00:22.330 --> 00:00:24.180
I want to graph this
linear equation.

00:00:24.180 --> 00:00:26.730
Before I even take out the graph
paper, what I could do

00:00:26.730 --> 00:00:28.450
is set up a table.

00:00:28.450 --> 00:00:31.840
Where I pick a bunch of x values
and then I can figure

00:00:31.840 --> 00:00:34.180
out what y value would
correspond to

00:00:34.180 --> 00:00:36.980
each of those x values.

00:00:36.980 --> 00:00:41.450
So for example, if x is equal
to-- let me start really low--

00:00:41.450 --> 00:00:45.670
if x is equal to minus 2--
or negative 2, I should

00:00:45.670 --> 00:00:47.310
say-- what is y?

00:00:47.310 --> 00:00:49.620
Well, you substitute
negative 2 up here.

00:00:49.620 --> 00:00:54.200
It would be 2 times
negative 2 plus 7.

00:00:54.200 --> 00:00:57.060
This is negative 4 plus 7.

00:00:57.060 --> 00:00:59.270
This is equal to 3.

00:00:59.270 --> 00:01:02.270
If x is equal to-- I'm just
picking x values at random

00:01:02.270 --> 00:01:05.209
that might be indicative of--
I'll probably do three or four

00:01:05.209 --> 00:01:06.190
points here.

00:01:06.190 --> 00:01:08.470
So what happens when
x is equal to 0?

00:01:08.470 --> 00:01:14.390
Then y is going to be equal
to 2 times 0 plus 7.

00:01:14.390 --> 00:01:16.530
Is going to be equal to 7.

00:01:19.410 --> 00:01:20.870
I just happen to be
going up by 2.

00:01:20.870 --> 00:01:22.270
You could be going up
by 1 or you could be

00:01:22.270 --> 00:01:23.840
picking numbers at random.

00:01:23.840 --> 00:01:27.600
When x is equal to
2, what is y?

00:01:27.600 --> 00:01:31.090
It'll be 2 times 2 plus 7.

00:01:31.090 --> 00:01:35.070
So 4 plus 7 is equal to 11.

00:01:35.070 --> 00:01:37.990
I could keep plotting
points if I like.

00:01:37.990 --> 00:01:39.630
We should already have
enough to graph it.

00:01:39.630 --> 00:01:43.700
Actually to plot any line, you
actually only need two points.

00:01:43.700 --> 00:01:45.130
So we already have one
more than necessary.

00:01:45.130 --> 00:01:46.530
Actually, let me just do one
more just to show you that

00:01:46.530 --> 00:01:48.050
this really is a line.

00:01:48.050 --> 00:01:50.750
So what happens when
x is equal to 4?

00:01:50.750 --> 00:01:54.670
Actually, just to not go up by
2, let's do x is equal to 8.

00:01:54.670 --> 00:01:56.600
Just to pick a random number.

00:01:56.600 --> 00:02:01.400
Then y is going to be 2 times 8
plus 7, which is-- well this

00:02:01.400 --> 00:02:05.010
might go off of our graph
paper-- but 2 times 8 is 16

00:02:05.010 --> 00:02:09.139
plus 7 is equal to 23.

00:02:09.139 --> 00:02:10.940
Now let's graph it.

00:02:10.940 --> 00:02:15.580
Let me do my y-axis
right there.

00:02:15.580 --> 00:02:18.050
That is my y-axis.

00:02:18.050 --> 00:02:20.550
Let me do my x-axis.

00:02:20.550 --> 00:02:23.220
I have a lot of positive values
here, so a lot of space

00:02:23.220 --> 00:02:25.540
on the positive y-side.

00:02:25.540 --> 00:02:27.200
That is my x-axis.

00:02:27.200 --> 00:02:30.540
And then I use the points x
is equal to negative 2.

00:02:30.540 --> 00:02:31.420
That's negative 1.

00:02:31.420 --> 00:02:42.440
That's 0, 1, 2, 3,
4, 5, 6, 7, 8.

00:02:42.440 --> 00:02:43.800
Those are our x values.

00:02:43.800 --> 00:02:45.430
Then we can go up
into the y-axis.

00:02:45.430 --> 00:02:47.190
I'll do it at a slightly
different scale because these

00:02:47.190 --> 00:02:49.630
numbers get large
very quickly.

00:02:49.630 --> 00:02:55.690
So maybe I'll do it in
increments of 2.

00:02:55.690 --> 00:03:06.800
So this could be 2, 4,
6, 8, 10, 12, 14, 16.

00:03:06.800 --> 00:03:08.370
I could just keep going
up there, but

00:03:08.370 --> 00:03:10.030
let's plot these points.

00:03:10.030 --> 00:03:13.540
So the first coordinate I have
is x is equal to negative 2, y

00:03:13.540 --> 00:03:15.770
is equal to 3.

00:03:15.770 --> 00:03:16.770
So I can write my coordinate.

00:03:16.770 --> 00:03:18.945
It's going to be the point
negative 2, 3.

00:03:18.945 --> 00:03:20.990
x is negative 2.

00:03:20.990 --> 00:03:22.320
y is 3.

00:03:22.320 --> 00:03:24.810
3 would land right over there.

00:03:24.810 --> 00:03:28.990
So that's our first one,
negative 2, 3.

00:03:28.990 --> 00:03:30.050
Then our next point.

00:03:30.050 --> 00:03:32.640
0, 7.

00:03:32.640 --> 00:03:33.600
We do it in that color.

00:03:33.600 --> 00:03:34.950
0, 7.

00:03:34.950 --> 00:03:35.880
x is 0.

00:03:35.880 --> 00:03:37.340
Y is 7.

00:03:37.340 --> 00:03:39.330
Right there.

00:03:39.330 --> 00:03:41.980
0, 7.

00:03:41.980 --> 00:03:44.200
We have this one
in green here.

00:03:44.200 --> 00:03:46.340
Point 2, 11.

00:03:46.340 --> 00:03:50.075
2, 11 would be right
about there.

00:03:53.000 --> 00:03:54.560
And then this last point-- this
is actually going to fall

00:03:54.560 --> 00:03:56.080
off of my graph.

00:03:56.080 --> 00:03:58.440
8, 23.

00:03:58.440 --> 00:04:02.410
That's going to be way
up here someplace.

00:04:02.410 --> 00:04:04.230
If you can even see
what I'm doing.

00:04:04.230 --> 00:04:06.650
This is 8, 23.

00:04:06.650 --> 00:04:09.090
If we connect the dots, you'll
see a line forms. Let me

00:04:09.090 --> 00:04:11.000
connect these dots.

00:04:11.000 --> 00:04:14.380
I've obviously hand drawn it, so
it might not be a perfectly

00:04:14.380 --> 00:04:15.390
straight line.

00:04:15.390 --> 00:04:17.290
If you had a computer do it, it
would be a straight line.

00:04:17.290 --> 00:04:20.029
So you could keep picking x
values and figuring out the

00:04:20.029 --> 00:04:21.920
corresponding y values.

00:04:21.920 --> 00:04:25.520
In the situation y is a function
of our x values.

00:04:25.520 --> 00:04:28.200
If you kept plotting every
point, you'll get every line.

00:04:28.200 --> 00:04:31.220
If you picked every possible x
and plotted every one, you get

00:04:31.220 --> 00:04:34.400
every point on the line.

00:04:34.400 --> 00:04:36.400
Let's do another problem.

00:04:36.400 --> 00:04:39.240
At the airport, you can
change your money from

00:04:39.240 --> 00:04:40.490
dollars into Euros.

00:04:43.850 --> 00:04:46.360
The service costs $5.

00:04:46.360 --> 00:04:49.380
and for every additional dollar,
you get EUR 0.7.

00:04:53.630 --> 00:04:57.470
Make a table for this and plot
the function on a graph.

00:04:57.470 --> 00:05:00.430
Use your graph to determine how
many Euros you would get

00:05:00.430 --> 00:05:03.320
if you give the office $50.

00:05:03.320 --> 00:05:09.380
I will write Euros is equal to--
so let's see, it's going

00:05:09.380 --> 00:05:10.710
to be dollars.

00:05:10.710 --> 00:05:12.040
So you're going to have
to give your dollars.

00:05:14.600 --> 00:05:19.290
Right off of the bat, they're
going to take $5.

00:05:19.290 --> 00:05:21.320
So dollars minus 5.

00:05:21.320 --> 00:05:24.430
So immediately this
service costs $5.

00:05:24.430 --> 00:05:26.480
And then everything that's
leftover-- this is your

00:05:26.480 --> 00:05:30.620
leftover-- you get EUR 0.7 for
every leftover dollars.

00:05:30.620 --> 00:05:34.410
You get 0.7 for whatever's
leftover.

00:05:34.410 --> 00:05:36.590
So this is the relationship.

00:05:36.590 --> 00:05:38.770
Now we can plot points-- we
could actually answer their

00:05:38.770 --> 00:05:39.750
question right off the bat.

00:05:39.750 --> 00:05:42.190
If you give them $50, we don't
even have to look at a graph.

00:05:42.190 --> 00:05:44.280
But we will look at a graph
right after this.

00:05:44.280 --> 00:05:48.190
So if you did Euros is equal
to-- if you have given them

00:05:48.190 --> 00:05:54.030
$50-- it would be 0.7
times 50 minus 5.

00:05:54.030 --> 00:05:55.350
You gave them 50.

00:05:55.350 --> 00:05:57.160
They took 5 as a service fee.

00:05:57.160 --> 00:06:04.000
So this is just $45 It would
be 0.7 times 45.

00:06:04.000 --> 00:06:05.060
I could do that right here.

00:06:05.060 --> 00:06:09.270
45 times 0.7.

00:06:09.270 --> 00:06:12.480
7 times 5 is 35.

00:06:12.480 --> 00:06:17.660
4 times 7 is 28 plus 3 is 31.

00:06:17.660 --> 00:06:19.800
And then we have only one number
behind the decimal,

00:06:19.800 --> 00:06:20.810
only this 7.

00:06:20.810 --> 00:06:22.880
So it's 31.5.

00:06:22.880 --> 00:06:29.530
So if you give them $50, you're
going to get EUR 31.5.

00:06:29.530 --> 00:06:31.300
Euros, not dollars.

00:06:31.300 --> 00:06:33.010
So we answered their question,
but let's actually do it

00:06:33.010 --> 00:06:34.610
graphically.

00:06:34.610 --> 00:06:37.330
Let's do a table.

00:06:37.330 --> 00:06:39.340
Maybe I'll get a
calculator out.

00:06:39.340 --> 00:06:40.580
I'll refer to that
in a little bit.

00:06:40.580 --> 00:06:42.230
So let's say dollars
you give them.

00:06:45.140 --> 00:06:49.200
And how many Euros do you get?

00:06:49.200 --> 00:06:51.810
I'll just put a bunch
of random numbers.

00:06:51.810 --> 00:06:54.100
If you give them $5, they're
just going to take

00:06:54.100 --> 00:06:55.580
your $5 for the fee.

00:06:55.580 --> 00:06:58.620
You're going to get $5 minus
5, which is 0 times 0.7.

00:06:58.620 --> 00:07:00.600
So you're going to
get nothing back.

00:07:00.600 --> 00:07:03.570
So there's really no good reason
for you to do that.

00:07:03.570 --> 00:07:06.520
Then if you give them $10.

00:07:06.520 --> 00:07:07.860
What's going to happen?

00:07:07.860 --> 00:07:13.630
If you give them $10, 10
minus 5 is 5 times 0.7.

00:07:13.630 --> 00:07:17.790
You're going to get $3-- or
I should say EUR 3.50.

00:07:17.790 --> 00:07:20.430
3.5 Euros, you'll get.

00:07:20.430 --> 00:07:26.690
Now what happens if
you give them $30?

00:07:26.690 --> 00:07:28.840
Actually let me say 25.

00:07:28.840 --> 00:07:33.420
If you give him $25,
25 minus 5 is 20.

00:07:33.420 --> 00:07:38.620
20 times 0.7 is $14.

00:07:38.620 --> 00:07:39.470
I'll do one more value.

00:07:39.470 --> 00:07:42.460
Let's say you gave them $55.

00:07:42.460 --> 00:07:43.650
This makes the math easy
because then you

00:07:43.650 --> 00:07:44.790
subtract that 5 out.

00:07:44.790 --> 00:07:53.170
55 minus 5 is 50 times
0.7 is $35.

00:07:53.170 --> 00:07:54.260
Is that right?

00:07:54.260 --> 00:07:56.230
Yep, that's right.

00:07:56.230 --> 00:07:57.890
You'll get EUR 35
I should say.

00:07:57.890 --> 00:07:58.500
These are all Euros.

00:07:58.500 --> 00:07:59.820
I keep wanting to say dollars.

00:07:59.820 --> 00:08:01.610
Let's plot this.

00:08:01.610 --> 00:08:03.620
All of these values are
positive, so I only have to

00:08:03.620 --> 00:08:05.240
draw the first quadrant here.

00:08:10.900 --> 00:08:17.540
And so the dollars-- let's go
in increments of 5, 10, 15,

00:08:17.540 --> 00:08:27.720
20, 25, 30, 35, 40,
45, 50, 55.

00:08:27.720 --> 00:08:30.070
I made my x-axis a little
shorter than I needed to.

00:08:30.070 --> 00:08:31.390
All the way up to 55.

00:08:31.390 --> 00:08:32.890
And then the y-axis.

00:08:32.890 --> 00:08:34.090
I'll go in increments of 5.

00:08:34.090 --> 00:08:45.550
So that's 5, 10, 15,
20, 25, 30, 35.

00:08:45.550 --> 00:08:47.330
Well that's a little bit too
much of an increment.

00:08:47.330 --> 00:08:49.220
35.

00:08:49.220 --> 00:08:50.230
Now let's plot these points.

00:08:50.230 --> 00:08:51.200
I give them $5.

00:08:51.200 --> 00:08:53.440
I get EUR 0.

00:08:53.440 --> 00:08:57.070
This right here is Euros.

00:08:57.070 --> 00:08:58.176
This is the dollars.

00:08:58.176 --> 00:09:00.930
The dollars is the independent
variable and we figure out the

00:09:00.930 --> 00:09:01.670
Euros from it.

00:09:01.670 --> 00:09:03.530
Or the Euros I get
is dependent on

00:09:03.530 --> 00:09:05.330
the dollars I get.

00:09:05.330 --> 00:09:07.790
If I give $10, I get EUR 3.50.

00:09:07.790 --> 00:09:09.430
3.50-- it's hard to read.

00:09:09.430 --> 00:09:11.850
Maybe 3.50 would be right
around there.

00:09:11.850 --> 00:09:14.040
If I give $25, I get EUR 14.

00:09:14.040 --> 00:09:17.650
25, 14 is right about there.

00:09:17.650 --> 00:09:19.090
Obviously, I'm hand drawing it,
so it's not going to be

00:09:19.090 --> 00:09:20.230
quite exact.

00:09:20.230 --> 00:09:22.460
If I get $55, I get EUR 35.

00:09:22.460 --> 00:09:26.900
So 55, 35 right there.

00:09:26.900 --> 00:09:30.450
If I were to connect to the
dots, I should get something

00:09:30.450 --> 00:09:32.410
that looks pretty
close to a line.

00:09:32.410 --> 00:09:34.340
If I did it-- if I was
a computer, it would

00:09:34.340 --> 00:09:38.150
be exactly a line.

00:09:38.150 --> 00:09:39.650
That looks pretty good.

00:09:39.650 --> 00:09:42.400
Then we could eyeball what
they asked us to do.

00:09:42.400 --> 00:09:44.750
Use your graph to determine how
many Euros you would get

00:09:44.750 --> 00:09:46.590
if you give the office $50.

00:09:46.590 --> 00:09:48.660
This is 50 right here.

00:09:48.660 --> 00:09:51.270
So you go bam, bam, bam, bam,
bam, bam, bam, bam.

00:09:51.270 --> 00:09:52.020
I'm at the graph.

00:09:52.020 --> 00:09:56.900
Then you go all the way--
actually I drew that last

00:09:56.900 --> 00:10:00.170
point on the graph a little
bit incorrectly.

00:10:00.170 --> 00:10:01.420
Let me.

00:10:04.660 --> 00:10:05.670
35 is right here.

00:10:05.670 --> 00:10:06.940
Let me redraw that point.

00:10:06.940 --> 00:10:10.870
35 is right there roughly.

00:10:10.870 --> 00:10:13.160
So 55, 35 is right there.

00:10:13.160 --> 00:10:14.900
So let me redraw my line.

00:10:14.900 --> 00:10:16.880
It will look-- I lost 25.

00:10:16.880 --> 00:10:20.470
25, 14 is right there.

00:10:20.470 --> 00:10:24.670
So my graph looks something
like that.

00:10:24.670 --> 00:10:26.010
That's my best attempt.

00:10:26.010 --> 00:10:27.350
Now let's answer the question.

00:10:27.350 --> 00:10:29.190
We give them $50 right there.

00:10:29.190 --> 00:10:32.480
You go up, up, up, up,
up, up, up. $50.

00:10:32.480 --> 00:10:34.030
The person is going to get.

00:10:34.030 --> 00:10:37.430
You go all the way to
the left-hand side.

00:10:37.430 --> 00:10:39.550
That's right about 31.50.

00:10:39.550 --> 00:10:41.370
We figured out exactly
using the formula.

00:10:41.370 --> 00:10:43.830
But you can see, you can eyeball
it from the graph and

00:10:43.830 --> 00:10:45.860
figure out any amount
of dollars.

00:10:45.860 --> 00:10:48.540
If you give them $20, you're
going to go all

00:10:48.540 --> 00:10:49.860
the way over here.

00:10:49.860 --> 00:10:53.090
You'll figure out that it should
be-- well $20 should be

00:10:53.090 --> 00:10:55.540
about 7.50.

00:10:55.540 --> 00:10:58.465
The imprecision in my graph-- in
my drawing the graph makes

00:10:58.465 --> 00:11:00.030
it a little bit less exact.

00:11:00.030 --> 00:11:03.330
When you say 20 minus 5 is 15.

00:11:03.330 --> 00:11:05.230
15 times-- actually it'll
be a little over

00:11:05.230 --> 00:11:06.640
$10, which is right.

00:11:06.640 --> 00:11:09.480
It's right over there.

00:11:09.480 --> 00:11:12.670
If you put $20 in there,
20 minus 5 is 15.

00:11:12.670 --> 00:11:18.340
15 times 0.7 is $10.50,
which is right there.

00:11:18.340 --> 00:11:19.850
So you can look at any point
in the graph and figure out

00:11:19.850 --> 00:11:22.250
how many Euros you'll get.

00:11:22.250 --> 00:11:24.130
Let's do this one where
we'll do a little bit

00:11:24.130 --> 00:11:26.270
of reading a graph.

00:11:26.270 --> 00:11:29.730
The graph-- I think it said
use the graph below.

00:11:29.730 --> 00:11:32.420
Oh, the graph below shows
a conversion chart for

00:11:32.420 --> 00:11:35.420
converting between weight
in kilograms

00:11:35.420 --> 00:11:36.990
and weight in pounds.

00:11:36.990 --> 00:11:38.710
Use it to convert the following
measurements.

00:11:38.710 --> 00:11:42.050
We have kilograms here
and pounds here.

00:11:42.050 --> 00:11:45.270
So they want 4 kilograms into
weight into pounds.

00:11:45.270 --> 00:11:47.820
So if we look at this
right here, 4

00:11:47.820 --> 00:11:52.640
kilograms is right there.

00:11:52.640 --> 00:11:55.150
We just follow where
the graph is.

00:11:55.150 --> 00:11:59.610
So 4 kilograms into pounds, it
looks like, I don't know, a

00:11:59.610 --> 00:12:02.210
little bit under 9 pounds.

00:12:02.210 --> 00:12:05.035
So a little bit less than--
so almost, I'll

00:12:05.035 --> 00:12:07.930
write almost 9 pounds.

00:12:07.930 --> 00:12:09.270
You can't exactly see.

00:12:09.270 --> 00:12:11.310
It's a little less than
9 pounds right there.

00:12:11.310 --> 00:12:15.370
4 kilograms. Now 9 kilograms.
We go over here.

00:12:15.370 --> 00:12:19.310
9 kilograms. Go all
the way up.

00:12:19.310 --> 00:12:24.700
That looks like almost
exactly 20 pounds.

00:12:24.700 --> 00:12:27.570
Here they say 12 pounds into
weight in kilograms. Actually

00:12:27.570 --> 00:12:30.030
kilograms is mass, but I
won't get particular.

00:12:30.030 --> 00:12:32.120
So 12 pounds.

00:12:32.120 --> 00:12:32.670
Go over here.

00:12:32.670 --> 00:12:33.460
Pounds.

00:12:33.460 --> 00:12:40.390
12 pounds in kilograms
looks like 5 1/2.

00:12:40.390 --> 00:12:43.090
Approximately 5 1/2.

00:12:43.090 --> 00:12:45.040
And then 17 pounds
to kilograms.

00:12:45.040 --> 00:12:47.390
So 17 is right there.

00:12:47.390 --> 00:12:58.120
17 pounds to kilograms looks
right about 7 1/2 kilograms.

00:12:58.120 --> 00:13:00.910
Anyway, hopefully that these
examples made you a little bit

00:13:00.910 --> 00:13:05.550
more comfortable with graphing
equations and reading graphs

00:13:05.550 --> 00:13:06.290
of equations.

00:13:06.290 --> 00:13:08.450
I'll see you in the
next video.