Comparing features of functions (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.630 --> 00:00:03.856
Which quadratic has the
lowest maximum value?

00:00:03.856 --> 00:00:06.230
So let's figure out the maximum
value for each of these--

00:00:06.230 --> 00:00:07.869
and they're defined
in different ways--

00:00:07.869 --> 00:00:09.410
and then see which
one is the lowest.

00:00:09.410 --> 00:00:11.600
And I'll start with the easiest.

00:00:11.600 --> 00:00:12.250
So h of x.

00:00:12.250 --> 00:00:15.022
We can just graphically look
at it, visually look at it,

00:00:15.022 --> 00:00:16.480
and say-- what's
the maximum point?

00:00:16.480 --> 00:00:19.560
And the maximum point looks
like it's right over here

00:00:19.560 --> 00:00:20.900
when x is equal to 4.

00:00:20.900 --> 00:00:26.110
And when x is equal to 4, y or
h of x is equal to negative 1.

00:00:26.110 --> 00:00:31.560
So the maximum for h of x
looks like it is negative 1.

00:00:31.560 --> 00:00:33.140
Now, what's the
maximum for g of x?

00:00:33.140 --> 00:00:34.860
And they've given us some
points here and here.

00:00:34.860 --> 00:00:36.530
Once again, we can just
eyeball it, and say-- well,

00:00:36.530 --> 00:00:38.400
what's the maximum
value they gave us?

00:00:38.400 --> 00:00:40.110
Well, 5 is the largest value.

00:00:40.110 --> 00:00:44.350
It happens when x is
equal to 0. g of 0 is 5.

00:00:44.350 --> 00:00:47.925
So the maximum value here is 5.

00:00:47.925 --> 00:00:48.860
Now, f of x.

00:00:48.860 --> 00:00:51.059
They just give us an
expression to define it.

00:00:51.059 --> 00:00:52.850
And so it's going take
a little bit of work

00:00:52.850 --> 00:00:54.990
to figure out what
the maximum value is.

00:00:54.990 --> 00:00:57.700
The easiest way to do
that for a quadratic

00:00:57.700 --> 00:00:59.920
is to complete the square.

00:00:59.920 --> 00:01:01.390
And so let's do it.

00:01:01.390 --> 00:01:05.940
So we have f of x is equal
to negative x squared

00:01:05.940 --> 00:01:08.855
plus 6x minus 1.

00:01:08.855 --> 00:01:10.480
I never like having
this negative here.

00:01:10.480 --> 00:01:11.730
So I'm going to factor it out.

00:01:11.730 --> 00:01:14.750
This is the same thing
as negative times

00:01:14.750 --> 00:01:18.670
x squared minus 6x and plus 1.

00:01:18.670 --> 00:01:20.420
And I'm going to write
the plus 1 out here

00:01:20.420 --> 00:01:23.122
because I'm fixing to
complete the square.

00:01:23.122 --> 00:01:25.080
Now, just as a review of
completing the square,

00:01:25.080 --> 00:01:27.450
we essentially want to add
and subtract the same number

00:01:27.450 --> 00:01:30.900
so that part of this
expression is a perfect square.

00:01:30.900 --> 00:01:33.420
And to figure out what number
we want to add and subtract,

00:01:33.420 --> 00:01:35.320
we look at the
coefficient on the x term.

00:01:35.320 --> 00:01:36.770
It's a negative 6.

00:01:36.770 --> 00:01:38.010
You take half of that.

00:01:38.010 --> 00:01:39.220
That's negative 3.

00:01:39.220 --> 00:01:40.710
And you square it.

00:01:40.710 --> 00:01:43.630
Negative 3 squared is 9.

00:01:43.630 --> 00:01:45.070
Now, we can't just add a 9.

00:01:45.070 --> 00:01:48.130
That would change the actual
value of the expression.

00:01:48.130 --> 00:01:50.502
We have to add a 9
and subtract a 9.

00:01:50.502 --> 00:01:52.960
And you might say-- well, why
are we adding and subtracting

00:01:52.960 --> 00:01:54.140
the same thing if
it doesn't change

00:01:54.140 --> 00:01:55.460
the value of the expression?

00:01:55.460 --> 00:01:56.980
And the whole
point is so that we

00:01:56.980 --> 00:01:59.720
can get this first
part of the expression

00:01:59.720 --> 00:02:01.610
to represent a perfect square.

00:02:01.610 --> 00:02:06.480
This x squared minus 6x
plus 9 is x minus 3 squared.

00:02:06.480 --> 00:02:10.479
So I can rewrite that
part as x minus 3

00:02:10.479 --> 00:02:17.380
squared and then minus 9--
or negative 9-- plus 1 is

00:02:17.380 --> 00:02:18.972
negative 8.

00:02:18.972 --> 00:02:20.430
Let me do that in
a different color

00:02:20.430 --> 00:02:22.410
so we can keep track of things.

00:02:22.410 --> 00:02:27.300
So this part right over
here is negative 8.

00:02:27.300 --> 00:02:31.970
And we still have the
negative out front.

00:02:31.970 --> 00:02:34.920
And so we can rewrite
this as-- if we distribute

00:02:34.920 --> 00:02:41.650
the negative sign-- negative
x minus 3 squared plus 8.

00:02:41.650 --> 00:02:43.920
Now, let's think about
what the maximum value is.

00:02:43.920 --> 00:02:45.420
And to understand
the maximum value,

00:02:45.420 --> 00:02:48.780
we have to interpret this
negative x minus 3 squared.

00:02:48.780 --> 00:02:53.100
Well, x minus 3 squared-- before
we think about the negative--

00:02:53.100 --> 00:02:56.980
that is always going
to be a positive value.

00:02:56.980 --> 00:02:58.646
Or it's always going
to be non-negative.

00:02:58.646 --> 00:03:00.104
But then, when we
make it negative,

00:03:00.104 --> 00:03:01.870
it's always going
to be non-positive.

00:03:01.870 --> 00:03:03.220
Think about it.

00:03:03.220 --> 00:03:05.780
If x is equal to 3, this
thing is going to be 0.

00:03:05.780 --> 00:03:08.500
And you take the negative
of that, it's going to be 0.

00:03:08.500 --> 00:03:11.360
x is anything else,
x is anything other

00:03:11.360 --> 00:03:13.450
than 3, this part
of the expression

00:03:13.450 --> 00:03:14.477
is going to be positive.

00:03:14.477 --> 00:03:15.810
But then, you have a minus sign.

00:03:15.810 --> 00:03:19.590
So you're going to subtract
that positive value from 8.

00:03:19.590 --> 00:03:25.610
So this actually
has a maximum value

00:03:25.610 --> 00:03:28.350
when this first term
right over here is 0.

00:03:28.350 --> 00:03:30.830
The only thing that this part
of the expression could do

00:03:30.830 --> 00:03:33.142
is subtract from the 8.

00:03:33.142 --> 00:03:34.600
If you want to get
a maximum value,

00:03:34.600 --> 00:03:36.070
this should be equal to 0.

00:03:36.070 --> 00:03:38.200
This equals 0 when
x is equal to 3.

00:03:38.200 --> 00:03:39.890
When x is equal to 3, this is 0.

00:03:39.890 --> 00:03:43.370
And our function hits
its maximum value of 8.

00:03:43.370 --> 00:03:45.750
So this has a max-- let
me do that in a color

00:03:45.750 --> 00:03:53.450
that you can actually read--
this has a max value of 8.

00:03:53.450 --> 00:03:56.180
So which has the
lowest maximum value?

00:03:56.180 --> 00:03:57.805
h of x.


Average Rate of Change Example 2)

WEBVTT
Kind: captions
Language: en

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The tortoise and
the hare are racing

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to see who can run 100 feet in
the shortest amount of time.

00:00:06.920 --> 00:00:10.690
Their positions in feet as a
function of time in seconds,

00:00:10.690 --> 00:00:13.820
T of t and H of t respectively,
are plotted below.

00:00:13.820 --> 00:00:17.500
So T of t is the position of the
tortoise as a function of time,

00:00:17.500 --> 00:00:18.600
and it's in feet.

00:00:18.600 --> 00:00:21.060
H of t is the
position of the hare.

00:00:21.060 --> 00:00:23.630
What is the meaning
of the rate of change

00:00:23.630 --> 00:00:28.530
of T of t on the interval
t is greater than 0

00:00:28.530 --> 00:00:29.750
and less than 20?

00:00:29.750 --> 00:00:33.280
So T of t, they were talking
about the tortoise here.

00:00:33.280 --> 00:00:35.530
This is the function T
of t right over here,

00:00:35.530 --> 00:00:36.880
this blue line.

00:00:36.880 --> 00:00:40.610
And it looks like it's moving
at a constant rate of change.

00:00:40.610 --> 00:00:43.090
The slope of this
line is constant.

00:00:43.090 --> 00:00:45.350
So let's see what this
rate of change is.

00:00:45.350 --> 00:00:47.470
If we start-- so
we could even start

00:00:47.470 --> 00:00:48.970
from the starting position.

00:00:48.970 --> 00:00:52.060
When 1 second goes
by, each notch here

00:00:52.060 --> 00:00:54.770
in the horizontal
axis is 1 second.

00:00:54.770 --> 00:00:57.440
You see 1, 2, 3, 4, 5.

00:00:57.440 --> 00:01:02.340
So every time 1 second
goes by, it looks like--

00:01:02.340 --> 00:01:04.129
and we have to be
very careful here.

00:01:04.129 --> 00:01:06.370
At first it looks like,
oh, maybe he moves 1 feet.

00:01:06.370 --> 00:01:09.140
But we have to be careful about
what these notches represent.

00:01:09.140 --> 00:01:14.278
It looks like these notches
represent 5, 10, 15, 20,

00:01:14.278 --> 00:01:15.420
25 feet.

00:01:15.420 --> 00:01:20.710
So every time 1 second goes
by, he moves forward 5 feet.

00:01:20.710 --> 00:01:24.150
Another second goes by,
he moves forward 5 feet.

00:01:24.150 --> 00:01:29.800
So it looks like the
rate of the tortoise,

00:01:29.800 --> 00:01:32.240
so I'll say the
tortoise rate, the rate

00:01:32.240 --> 00:01:35.770
of the change of distance
with respect to time

00:01:35.770 --> 00:01:40.650
is 5 feet for every second.

00:01:40.650 --> 00:01:43.290
So let's see which of our
choices seem to match that.

00:01:43.290 --> 00:01:44.970
And that is over
the entire interval.

00:01:44.970 --> 00:01:46.550
The interval starts
here, this is 0,

00:01:46.550 --> 00:01:48.090
and 20 is right over here.

00:01:48.090 --> 00:01:49.840
And it looks like we
have a constant rate,

00:01:49.840 --> 00:01:51.390
we have a constant slope.

00:01:51.390 --> 00:01:53.100
We have the same
inclination of this line

00:01:53.100 --> 00:01:57.840
the entire time all the
way until t is equal to 20.

00:01:57.840 --> 00:01:59.880
So this first choice
says the tortoise walks

00:01:59.880 --> 00:02:04.000
at 5 feet per second the whole
time, and this looks right.

00:02:04.000 --> 00:02:06.900
Our tortoise has a constant
rate of 5 feet per second.

00:02:06.900 --> 00:02:09.550
This walks might be
an understatement.

00:02:09.550 --> 00:02:12.410
5 feet per second is a
pretty good speed, especially

00:02:12.410 --> 00:02:13.510
for a tortoise.

00:02:13.510 --> 00:02:16.364
But we are going to
go with that one.

00:02:16.364 --> 00:02:18.280
Now, let's just look at
the other ones for fun

00:02:18.280 --> 00:02:20.970
to see how true they are
or see if they could have

00:02:20.970 --> 00:02:24.030
somehow applied to this problem.

00:02:24.030 --> 00:02:27.490
The hare stops moving
after 5 seconds.

00:02:27.490 --> 00:02:30.580
So let's see, this is
time 0, this is 5 seconds.

00:02:30.580 --> 00:02:35.820
So if we look at the hare's
position after 5 seconds,

00:02:35.820 --> 00:02:39.360
so after time
equals 5 seconds, it

00:02:39.360 --> 00:02:42.270
does indeed look like the hare's
position-- the hare is just

00:02:42.270 --> 00:02:46.230
sitting there at the
90-foot marker line

00:02:46.230 --> 00:02:47.640
and is not moving at all.

00:02:47.640 --> 00:02:50.450
The rate of change of
position with respect

00:02:50.450 --> 00:02:53.510
to time for the hare
at that point is 0.

00:02:53.510 --> 00:02:56.561
So this does seem true, it just
doesn't apply to the question.

00:02:56.561 --> 00:02:58.310
It's not the meaning
of the rate of change

00:02:58.310 --> 00:02:59.940
of T of t on this interval.

00:02:59.940 --> 00:03:01.940
This is applying to H of t.

00:03:01.940 --> 00:03:05.340
The hare runs at a speed of 18
feet per second for the first 5

00:03:05.340 --> 00:03:06.240
seconds.

00:03:06.240 --> 00:03:07.860
So let's see if this is true.

00:03:07.860 --> 00:03:11.330
For the first 5 seconds, so
this is 5 seconds going by,

00:03:11.330 --> 00:03:18.570
the hare is able to cover
a distance of 90 feet.

00:03:18.570 --> 00:03:23.130
So the hare is able to
go 90 feet in 5 seconds,

00:03:23.130 --> 00:03:28.100
so for the first 5
seconds, the hare

00:03:28.100 --> 00:03:33.030
is-- hare's rate is 90 feet.

00:03:33.030 --> 00:03:37.240
You could say average rate of
change is 90 feet in 5 seconds.

00:03:37.240 --> 00:03:43.850
And 90 divided by 5 is
indeed 18 feet per second.

00:03:43.850 --> 00:03:46.480
So once again, this
third statement is true,

00:03:46.480 --> 00:03:48.240
it just doesn't apply to T of t.

00:03:48.240 --> 00:03:50.530
It's not addressing
the question.

00:03:50.530 --> 00:03:53.706
Now this fourth choice is
that the hare wins the race.

00:03:53.706 --> 00:03:55.330
Now, winning the
race, they have to get

00:03:55.330 --> 00:03:57.370
to 100 feet in the
shortest amount of time.

00:03:57.370 --> 00:03:59.550
This is 100 feet
right over here.

00:03:59.550 --> 00:04:00.650
So let's see.

00:04:00.650 --> 00:04:02.980
So it looks like only
the tortoise even

00:04:02.980 --> 00:04:03.900
gets to that point.

00:04:03.900 --> 00:04:07.760
At 20 seconds, the tortoise
gets to the 100-foot mark.

00:04:07.760 --> 00:04:10.280
The hare is still hanging
around taking a nap or something

00:04:10.280 --> 00:04:11.560
at 90 feet.

00:04:11.560 --> 00:04:13.705
So this is actually
outright false.

00:04:13.705 --> 00:04:15.230
The tortoise wins the race.

00:04:15.230 --> 00:04:18.680
So we will definitely go
with our first choice.


Evaluating functions given their formula

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:03.190
The function f of
x is defined as f

00:00:03.190 --> 00:00:06.690
of x is equal to
49 minus x squared.

00:00:06.690 --> 00:00:09.179
Find the value of f of 5.

00:00:09.179 --> 00:00:10.970
So whenever you're
dealing with a function,

00:00:10.970 --> 00:00:11.870
you take your input.

00:00:11.870 --> 00:00:14.620
In this case, our input
is going to be our 5.

00:00:14.620 --> 00:00:17.340
We input it into our
little function box,

00:00:17.340 --> 00:00:20.360
and we need to get our output.

00:00:20.360 --> 00:00:22.670
And they defined the
function box here

00:00:22.670 --> 00:00:25.900
as whatever your input is,
take that, square it, and then

00:00:25.900 --> 00:00:28.230
subtract it from 49.

00:00:28.230 --> 00:00:33.470
So f of 5, every time I
see an x here, since f of x

00:00:33.470 --> 00:00:35.720
is equal to this,
every time I see an x,

00:00:35.720 --> 00:00:37.320
I would replace
it with the input.

00:00:37.320 --> 00:00:40.880
So f of 5 is going
to be equal to 49

00:00:40.880 --> 00:00:43.030
minus-- instead of
writing x squared,

00:00:43.030 --> 00:00:45.210
I would write 5 squared.

00:00:45.210 --> 00:00:49.800
So this is equal to 49 minus 25.

00:00:49.800 --> 00:00:55.740
And 49 minus 25 is equal to 24.

00:00:55.740 --> 00:00:58.090
And we are done.


How to find the average rate of change from a table

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:02.700
What is the average
rate of change

00:00:02.700 --> 00:00:05.890
of y of x over the
interval negative 5

00:00:05.890 --> 00:00:09.330
is less than x is
less than negative 2?

00:00:09.330 --> 00:00:11.960
So this is x is
equal to negative 5.

00:00:11.960 --> 00:00:16.750
When x is equal to negative
5, y of x is equal to 6.

00:00:16.750 --> 00:00:23.180
And when x is equal to negative
2, y of x is equal to 0.

00:00:23.180 --> 00:00:25.890
So to figure out the
average rate of change,

00:00:25.890 --> 00:00:32.980
so the average rate
of change, of y

00:00:32.980 --> 00:00:35.720
of x, with respect--
and we can assume it's

00:00:35.720 --> 00:00:46.750
with respect to x-- let me
make that a little bit neater--

00:00:46.750 --> 00:00:49.370
this is going to
be the change in y

00:00:49.370 --> 00:00:53.480
of x over that interval over the
change of x of that interval.

00:00:53.480 --> 00:00:57.810
And the shorthand for change
is this triangle symbol, delta.

00:00:57.810 --> 00:01:00.110
Delta y-- I'll just write y.

00:01:00.110 --> 00:01:01.470
I could write delta y of x.

00:01:01.470 --> 00:01:02.310
It's delta y.

00:01:02.310 --> 00:01:05.620
Change in y over
our change in x.

00:01:05.620 --> 00:01:07.780
That's going to be our
average rate of change

00:01:07.780 --> 00:01:09.070
over this interval.

00:01:09.070 --> 00:01:12.200
So how much did y change
over this interval?

00:01:12.200 --> 00:01:15.240
So y went from a 6 to a 0.

00:01:15.240 --> 00:01:17.200
So let's say that we
can kind of view this

00:01:17.200 --> 00:01:18.850
as our endpoint right over here.

00:01:18.850 --> 00:01:20.370
So this is our end.

00:01:20.370 --> 00:01:21.862
This is our start.

00:01:21.862 --> 00:01:23.820
And we could have done
it the other way around.

00:01:23.820 --> 00:01:25.194
We would get a
consistent result.

00:01:25.194 --> 00:01:26.940
But since this is
higher up on the list,

00:01:26.940 --> 00:01:28.190
let's call this the start.

00:01:28.190 --> 00:01:29.527
And the x is a lower value.

00:01:29.527 --> 00:01:30.610
We'll call that our start.

00:01:30.610 --> 00:01:31.900
This is our end.

00:01:31.900 --> 00:01:33.570
So we start at 6.

00:01:33.570 --> 00:01:34.920
We end at 0.

00:01:34.920 --> 00:01:38.150
So our change in y is
going to be negative 6.

00:01:38.150 --> 00:01:41.490
We went down by 6
in the y direction.

00:01:41.490 --> 00:01:42.310
It's negative 6.

00:01:42.310 --> 00:01:44.940
You could say that's 0 minus 6.

00:01:44.940 --> 00:01:47.240
And our change in x, well,
we are at negative 5,

00:01:47.240 --> 00:01:49.190
and we go up to negative 2.

00:01:49.190 --> 00:01:51.270
We increased by 3.

00:01:54.330 --> 00:01:59.710
So when we increased x by
3, we decreased y of x by 6.

00:01:59.710 --> 00:02:01.820
Or if we want to simplify
this right over here,

00:02:01.820 --> 00:02:05.220
negative 6 over 3 is the
same thing as negative 2.

00:02:05.220 --> 00:02:06.920
So our average
rate of change of y

00:02:06.920 --> 00:02:10.350
of x over the interval from
negative 5 to negative 2

00:02:10.350 --> 00:02:12.430
is negative 2.

00:02:12.430 --> 00:02:15.340
Every time, on
average, x increased 1,

00:02:15.340 --> 00:02:19.570
y went down by negative 2.


Evaluating functions given their graph

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:02.840
The function f of x is graphed.

00:00:02.840 --> 00:00:05.210
Find f of negative 1.

00:00:05.210 --> 00:00:07.660
So this graph right
over here is essentially

00:00:07.660 --> 00:00:09.090
a definition of our function.

00:00:09.090 --> 00:00:12.360
It tells us, given the allowed
inputs into our function,

00:00:12.360 --> 00:00:14.300
what would the function output?

00:00:14.300 --> 00:00:16.079
So here, they're
saying, look, what

00:00:16.079 --> 00:00:21.270
gets output when we input
x is equal to negative 1?

00:00:21.270 --> 00:00:24.340
So x equals negative 1
is right over here. x

00:00:24.340 --> 00:00:25.730
is equal to negative 1.

00:00:25.730 --> 00:00:30.040
And our function
graph is right at 6

00:00:30.040 --> 00:00:32.340
when f is equal to negative 1.

00:00:32.340 --> 00:00:36.910
So we can say that f of
negative 1 is equal to 6.

00:00:36.910 --> 00:00:42.665
Let me write that over here.
f of negative 1 is equal to 6.


Linear and nonlinear functions (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:04.720
Luis and Kate have two video
games they want to play.

00:00:04.720 --> 00:00:10.620
They plan to spend exactly 45
minutes playing the two games.

00:00:10.620 --> 00:00:12.900
They want to use an
equation to express

00:00:12.900 --> 00:00:16.010
the relationship between the
number of minutes they spend

00:00:16.010 --> 00:00:19.140
playing Super Bologna Man
and the number of minutes

00:00:19.140 --> 00:00:23.550
they spend playing You
Have to Cut the Wire.

00:00:23.550 --> 00:00:25.870
Can this relationship
be represented

00:00:25.870 --> 00:00:28.370
using a linear equation?

00:00:28.370 --> 00:00:32.299
So let's see if it can.

00:00:32.299 --> 00:00:35.220
Let's define one variable
for the amount of time,

00:00:35.220 --> 00:00:41.380
the number of minutes they
spend playing Super Bologna Man.

00:00:41.380 --> 00:00:44.010
Let's define that as, well
let's just say that's x.

00:00:44.010 --> 00:00:50.420
So x is equal to time
playing Bologna Man.

00:00:50.420 --> 00:00:53.700
I'll write Bologna right here.

00:00:53.700 --> 00:01:00.870
And let's define y as y is
equal to the number of minutes

00:01:00.870 --> 00:01:04.340
they spend playing You
Have to Cut the Wire.

00:01:04.340 --> 00:01:14.592
So it's time playing, I'll
just call it Wire for short.

00:01:14.592 --> 00:01:16.050
So if we have the
minutes they play

00:01:16.050 --> 00:01:19.660
time playing Bologna and the
time playing Cut Your Wire.

00:01:19.660 --> 00:01:23.970
If I were to add those two
together, so if I say x plus y.

00:01:23.970 --> 00:01:28.420
I'll write that plus in a
neutral color, x plus y.

00:01:28.420 --> 00:01:31.640
What does this need
to be equal to?

00:01:31.640 --> 00:01:33.970
The time I play Bologna
Man plus the time

00:01:33.970 --> 00:01:35.820
I play Have to Cut the Wire.

00:01:35.820 --> 00:01:37.610
Well if I add them
together, they

00:01:37.610 --> 00:01:41.230
want to spend exactly 45
minutes playing both games.

00:01:41.230 --> 00:01:45.760
So this is going to be
equal to 45 minutes.

00:01:45.760 --> 00:01:50.130
Now we have set up
an equation that

00:01:50.130 --> 00:01:53.360
relates the time playing
Bologna Man and the time

00:01:53.360 --> 00:01:56.730
playing You Have
to Cut the Wire.

00:01:56.730 --> 00:02:01.010
But now we have to think about
is this a linear relationship?

00:02:01.010 --> 00:02:04.370
So one way to think about
it is the real giveaway

00:02:04.370 --> 00:02:06.490
for a linear relationship
is if you can write it

00:02:06.490 --> 00:02:08.750
in the traditional
form of a line.

00:02:08.750 --> 00:02:10.360
So if you can write
it in the y is

00:02:10.360 --> 00:02:15.590
equal to mx plus b form, where
m is the slope of the line

00:02:15.590 --> 00:02:17.920
and b is the y-intercept.

00:02:17.920 --> 00:02:19.980
So let's see if we can do that.

00:02:19.980 --> 00:02:21.850
Well if we want to
do that here, we

00:02:21.850 --> 00:02:24.010
could just subtract
x from both sides.

00:02:24.010 --> 00:02:26.600
You subtract x from
both sides, you

00:02:26.600 --> 00:02:28.750
get-- so let's
subtract an x over

00:02:28.750 --> 00:02:31.660
here, let's subtract
an x over here.

00:02:31.660 --> 00:02:34.320
So negative x plus this and
then subtract an x there.

00:02:34.320 --> 00:02:36.000
Well, that's going
to cancel, and you're

00:02:36.000 --> 00:02:37.930
going to be left
with-- and I'm going

00:02:37.930 --> 00:02:40.240
try to write it in this
form right over here--

00:02:40.240 --> 00:02:44.727
y is equal to 45.

00:02:44.727 --> 00:02:46.310
Let me do that in
the same color, just

00:02:46.310 --> 00:02:48.820
to make it not be confusing.

00:02:48.820 --> 00:02:53.500
y is equal to, and I'll
write the negative x first

00:02:53.500 --> 00:02:55.750
because we have the x term
right over here first.

00:02:55.750 --> 00:03:01.335
So y is equal to
negative x plus 45.

00:03:04.350 --> 00:03:06.910
All I did is I switched
these two terms around.

00:03:06.910 --> 00:03:09.370
But you see here,
it has that form.

00:03:09.370 --> 00:03:11.000
And you might say
wait what is m here?

00:03:11.000 --> 00:03:12.900
I see that b is 45.

00:03:12.900 --> 00:03:14.540
Well if I write
negative x, that's

00:03:14.540 --> 00:03:18.940
the same thing as
writing negative 1x.

00:03:18.940 --> 00:03:20.730
So this is definitely a line.

00:03:20.730 --> 00:03:24.350
I was able to write it in
this form right over here.

00:03:24.350 --> 00:03:27.470
So can this relationship
be represented

00:03:27.470 --> 00:03:29.290
using a linear equation?

00:03:29.290 --> 00:03:33.611
Absolutely, absolutely yes.


Function notation in context example

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:03.310
Frank wants to fill
up a spherical water

00:00:03.310 --> 00:00:05.800
balloon with as much
water as possible.

00:00:05.800 --> 00:00:07.710
The balloons he
bought can stretch

00:00:07.710 --> 00:00:11.940
to a radius of 3
inches-- not too big.

00:00:11.940 --> 00:00:15.490
If the volume of a sphere
is-- and this is volume

00:00:15.490 --> 00:00:20.290
as a function of radius--
is equal to 4/3 pi r cubed,

00:00:20.290 --> 00:00:23.320
what volume of water
in cubic inches

00:00:23.320 --> 00:00:25.580
can Frank put into the balloon?

00:00:25.580 --> 00:00:27.340
So this function
definition is going--

00:00:27.340 --> 00:00:29.700
if you give it a
radius in inches,

00:00:29.700 --> 00:00:32.280
it's going to produce a
volume in cubic inches.

00:00:32.280 --> 00:00:33.470
So let's rewrite it.

00:00:33.470 --> 00:00:42.270
Volume as a function of radius
is equal to 4/3 pi r cubed.

00:00:42.270 --> 00:00:43.910
Now, they say the
balloons he bought

00:00:43.910 --> 00:00:46.680
can stretch to a
radius of 3 inches.

00:00:46.680 --> 00:00:49.670
So let's think about, if the
radius gets to 3 inches, what

00:00:49.670 --> 00:00:52.110
the volume of that
balloon is going to be.

00:00:52.110 --> 00:00:55.030
So we essentially would
just input 3 inches

00:00:55.030 --> 00:00:57.770
into our function definition.

00:00:57.770 --> 00:01:01.940
So everywhere where we see an
r, we would replace it with a 3.

00:01:01.940 --> 00:01:03.730
So we could write--
and just to be clear,

00:01:03.730 --> 00:01:05.230
let me rewrite it
in the same color.

00:01:05.230 --> 00:01:07.400
V of-- that's not
the same color.

00:01:07.400 --> 00:01:12.400
We do it in that brownish
color right over here.

00:01:12.400 --> 00:01:23.890
So V of 3 is equal to 4/3
pi-- and instead of r cubed,

00:01:23.890 --> 00:01:31.640
I would write 3
cubed-- 4/3 pi 3 cubed.

00:01:31.640 --> 00:01:33.390
This is how the function
definition works.

00:01:33.390 --> 00:01:35.620
Whatever we input
here, it will replace

00:01:35.620 --> 00:01:37.830
the r in the expression.

00:01:37.830 --> 00:01:41.070
So V of 3 is going to
be equal to-- so this

00:01:41.070 --> 00:01:46.450
is going to be equal to 4/3 pi
times-- 3 to the third power

00:01:46.450 --> 00:01:48.030
is 27.

00:01:48.030 --> 00:01:53.130
27 divided by 3 is
9, so this is 9.

00:01:53.130 --> 00:01:55.890
9 times 4 is 36 pi.

00:01:55.890 --> 00:01:58.170
So this is equal to 36 pi.

00:01:58.170 --> 00:02:00.590
And since this was
in inches, our volume

00:02:00.590 --> 00:02:06.510
is going to be in inches
cubed or cubic inches.

00:02:06.510 --> 00:02:08.900
So that's the volume
of water that Frank

00:02:08.900 --> 00:02:12.750
can put in the balloon--
36 pi cubic inches.


Linear and nonlinear functions (example 3)

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:04.350
Find the missing value to
make the table represent

00:00:04.350 --> 00:00:06.600
a linear equation.

00:00:06.600 --> 00:00:08.560
So let's see this
table right over here.

00:00:08.560 --> 00:00:12.000
So when x is equal
to 1, y is 3/2.

00:00:12.000 --> 00:00:15.035
When x is 2, y is equal to 3.

00:00:15.035 --> 00:00:16.160
So let's see what happened.

00:00:16.160 --> 00:00:22.020
When x increased by
1, what did y do?

00:00:22.020 --> 00:00:25.400
Well looks like y
increased by 3 and 1/2

00:00:25.400 --> 00:00:27.000
is the same thing as 1 and 1/2.

00:00:27.000 --> 00:00:31.220
So to go from 1 and 1/2 to
3, it increased by 1 and 1/2,

00:00:31.220 --> 00:00:34.060
or you could say it
increased by 3/2.

00:00:34.060 --> 00:00:38.290
You could say that 3 is the
same thing as 6/2, 6/2 minus 3/2

00:00:38.290 --> 00:00:40.360
is another 3/2.

00:00:40.360 --> 00:00:40.870
All right.

00:00:40.870 --> 00:00:43.220
Now when we go
from 2 to 3, we're

00:00:43.220 --> 00:00:46.400
increasing by 1 again in x.

00:00:46.400 --> 00:00:47.960
And what are we doing in y?

00:00:47.960 --> 00:00:52.440
So we're going from 3, which is
the same thing as 6/2 to 9/2.

00:00:52.440 --> 00:00:56.830
So once again, we are
increasing by 3/2.

00:00:56.830 --> 00:00:59.500
So in order for this
to be a linear equation

00:00:59.500 --> 00:01:02.340
or a linear relationship,
every time we

00:01:02.340 --> 00:01:05.810
increase by 1 in
the x direction,

00:01:05.810 --> 00:01:08.090
we need to increase by 3/2.

00:01:08.090 --> 00:01:12.980
If we increase by 2, we need
to increase by 2 times 3/2.

00:01:12.980 --> 00:01:15.064
So what are we doing over
here on this fourth term

00:01:15.064 --> 00:01:15.605
on the table?

00:01:15.605 --> 00:01:16.540
Well we're increasing.

00:01:16.540 --> 00:01:21.200
We're going from 3 to 8
so we are increasing by 5.

00:01:21.200 --> 00:01:23.990
So if we're increasing
x by 5, then we

00:01:23.990 --> 00:01:34.890
need to increase y by 5
times 3/2 or 15 over 2.

00:01:34.890 --> 00:01:39.820
So that's the amount that
we have to increase y by.

00:01:39.820 --> 00:01:44.470
If we started at 9/2 and we're
going to increase by 15/2,

00:01:44.470 --> 00:01:48.480
so it's going to
be 9/2 plus 15/2,

00:01:48.480 --> 00:01:50.070
this is how much
we increment by.

00:01:50.070 --> 00:01:53.140
Remember, we increment
3/2 every time x moves 1.

00:01:53.140 --> 00:01:54.800
This time, x moved 5.

00:01:54.800 --> 00:01:57.010
So we're incrementing
by 15/2, or you

00:01:57.010 --> 00:02:00.100
could say we're incrementing
by 3/2 five times.

00:02:00.100 --> 00:02:02.020
But this is going
to be equal to 9

00:02:02.020 --> 00:02:08.120
plus 15 is 24 over 2
which is equal to 12.

00:02:08.120 --> 00:02:12.930
And so in the box, we could
write 12, and we are done.


Linear and nonlinear functions (example 1)

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:05.134
Does the following table
represent a linear equation?

00:00:05.134 --> 00:00:06.550
So let's see what's
going on here.

00:00:06.550 --> 00:00:09.360
When x is negative 7, y is 4.

00:00:09.360 --> 00:00:11.900
Then when x is
negative 3, y is 3.

00:00:11.900 --> 00:00:14.960
So let's see what happened
to what our change in x was.

00:00:14.960 --> 00:00:16.880
So our change in
x-- and I could even

00:00:16.880 --> 00:00:19.080
write it over here,
our change in x.

00:00:19.080 --> 00:00:22.050
So going from negative
7 to negative 3,

00:00:22.050 --> 00:00:24.610
we had an increase in 4 in x.

00:00:24.610 --> 00:00:26.510
And what was our change in y?

00:00:26.510 --> 00:00:29.000
And this triangle, that's
just the Greek letter delta.

00:00:29.000 --> 00:00:31.450
It's shorthand for "change in."

00:00:31.450 --> 00:00:34.380
Well, our change in y
when x increased by 4,

00:00:34.380 --> 00:00:36.680
our y-value went from 4 to 3.

00:00:36.680 --> 00:00:39.470
So our change in
y is negative 1.

00:00:39.470 --> 00:00:42.870
Now, in order for this
to be a linear equation,

00:00:42.870 --> 00:00:45.580
the ratio between our change
in y and our change in x

00:00:45.580 --> 00:00:47.370
has to be constant.

00:00:47.370 --> 00:00:50.780
So our change in
y over change in x

00:00:50.780 --> 00:00:53.210
for any two points
in this equation

00:00:53.210 --> 00:00:56.730
or any two points in the table
has to be the same constant.

00:00:56.730 --> 00:00:59.930
When x changed by 4, y
changed by negative 1.

00:00:59.930 --> 00:01:03.100
Or when y changed by
negative 1, x changed by 4.

00:01:03.100 --> 00:01:06.310
So we have to have a
constant change in y

00:01:06.310 --> 00:01:09.300
with respect to x
of negative 1/4.

00:01:09.300 --> 00:01:11.210
Let's see if this is true.

00:01:11.210 --> 00:01:13.900
So the next two points, when
I go from negative 3 to 1,

00:01:13.900 --> 00:01:16.520
once again I'm
increasing x by 4.

00:01:16.520 --> 00:01:19.520
And once again, I'm
decreasing y by negative 1.

00:01:19.520 --> 00:01:22.110
So we have that same ratio.

00:01:22.110 --> 00:01:24.270
Now, let's look at
this last point.

00:01:24.270 --> 00:01:27.670
When we go from 1 to
7 in the x-direction,

00:01:27.670 --> 00:01:29.640
we are increasing by 6.

00:01:29.640 --> 00:01:33.390
And when we go from 2 to 1,
we are still decreasing by 1.

00:01:33.390 --> 00:01:37.140
So now this ratio, going
from this third point

00:01:37.140 --> 00:01:40.940
to this fourth point,
is negative 1/6.

00:01:40.940 --> 00:01:41.800
So it is not.

00:01:41.800 --> 00:01:44.840
So just for this last
point right over here,

00:01:44.840 --> 00:01:48.660
for this last point, our change
in y over change in x, or I

00:01:48.660 --> 00:01:50.710
should say, really, between
these last two points

00:01:50.710 --> 00:01:54.710
right over here, our change
in y over change in x--

00:01:54.710 --> 00:01:57.036
let me clear this up.

00:01:57.036 --> 00:01:57.910
Let me make it clear.

00:01:57.910 --> 00:02:01.500
So just between these
last-- in magenta.

00:02:01.500 --> 00:02:04.080
Just between these last
two points over here,

00:02:04.080 --> 00:02:09.410
our change in y is negative
1, and our change in x is 6.

00:02:09.410 --> 00:02:13.620
So we have a different rate of
change of y with respect to x.

00:02:13.620 --> 00:02:15.880
Because we had a different
rate of change of y

00:02:15.880 --> 00:02:18.940
with respect to x, or ratio
between our change in y

00:02:18.940 --> 00:02:22.560
and change in x, this is
not a linear equation.

00:02:22.560 --> 00:02:25.490
No, not a linear equation.


Comparing features of functions (example 3)

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:04.210
Which of the features are
shared by f of x and g of x?

00:00:04.210 --> 00:00:06.030
Select all that apply.

00:00:06.030 --> 00:00:07.760
So they give us f of
x as being defined

00:00:07.760 --> 00:00:09.620
as x to the third minus x.

00:00:09.620 --> 00:00:12.800
And they define g of x,
essentially, with this graph.

00:00:12.800 --> 00:00:14.220
So what are our options?

00:00:14.220 --> 00:00:17.020
So the first one is
that they are both odd.

00:00:17.020 --> 00:00:19.080
So just by looking
at g of x, you

00:00:19.080 --> 00:00:20.930
can tell that it is not odd.

00:00:20.930 --> 00:00:23.230
The biggest giveaway
is that an odd function

00:00:23.230 --> 00:00:26.600
would go through the origin.

00:00:26.600 --> 00:00:29.507
G of zero would have
to be equal to zero.

00:00:29.507 --> 00:00:31.340
If you want to go
straight to the definition

00:00:31.340 --> 00:00:34.390
of an odd function,
g of x would have

00:00:34.390 --> 00:00:38.330
to be equal to the negative
of g of negative x.

00:00:38.330 --> 00:00:43.560
So for example, g of
3 looks like it is 4.

00:00:43.560 --> 00:00:45.720
g of 3 is equal to 4.

00:00:49.760 --> 00:00:52.410
In order for it to be
odd, g of negative 3

00:00:52.410 --> 00:00:55.080
would have to be
equal to negative 4.

00:00:55.080 --> 00:00:59.320
But we see that g of negative
3 is not equal to negative 4.

00:00:59.320 --> 00:01:01.037
So this one is
definitely not odd.

00:01:01.037 --> 00:01:02.370
So this statement can't be true.

00:01:02.370 --> 00:01:03.860
They both can't be odd.

00:01:03.860 --> 00:01:05.250
So that's not right.

00:01:05.250 --> 00:01:08.040
They share an x-intercept.

00:01:08.040 --> 00:01:10.310
So g of x only has
one x-intercept.

00:01:10.310 --> 00:01:12.220
It intersects the
x-axis right over here

00:01:12.220 --> 00:01:14.670
at x equals negative 3.

00:01:14.670 --> 00:01:17.650
Now let's think about the
x-intercepts of f of x.

00:01:17.650 --> 00:01:20.270
And to do that, we just need
to factor this expression,

00:01:20.270 --> 00:01:23.740
f of x is equal to x to
the third minus x, which

00:01:23.740 --> 00:01:25.440
is the same thing
if we factor an x

00:01:25.440 --> 00:01:29.700
out of x times x
squared minus 1.

00:01:29.700 --> 00:01:31.835
X squared minus 1 is the
difference of squares.

00:01:31.835 --> 00:01:34.210
So we could rewrite that as--
so we'll write our x first,

00:01:34.210 --> 00:01:35.080
this x.

00:01:35.080 --> 00:01:40.710
And then x squared minus 1
is x plus 1 times x minus 1.

00:01:40.710 --> 00:01:42.800
So when does f of x equal 0?

00:01:42.800 --> 00:01:46.830
Well, f of x is equal to
0 when x is equal to 0.

00:01:46.830 --> 00:01:50.680
When x is equal to 0, that would
make this entire expression 0.

00:01:50.680 --> 00:01:52.910
When x is equal to
negative 1, that

00:01:52.910 --> 00:01:56.490
would make this term, and
thus the entire expression, 0.

00:01:56.490 --> 00:01:59.410
And when x is equal
to positive 1,

00:01:59.410 --> 00:02:01.260
that would make
this last part zero,

00:02:01.260 --> 00:02:03.390
which would make this
entire product 0.

00:02:03.390 --> 00:02:06.430
So here are the zeroes
of f of x, and none

00:02:06.430 --> 00:02:08.949
of these coincide with
the zeroes of g of x.

00:02:08.949 --> 00:02:12.820
So they don't share
an x-intercept.

00:02:12.820 --> 00:02:16.570
They have the same end behavior.

00:02:16.570 --> 00:02:17.810
Now this is interesting.

00:02:17.810 --> 00:02:21.820
This is saying what's
happening as x gets really,

00:02:21.820 --> 00:02:25.720
really, really, really large,
or as x gets really, really,

00:02:25.720 --> 00:02:27.241
really, really, really small.

00:02:27.241 --> 00:02:29.240
So we could just think
about it right over here.

00:02:29.240 --> 00:02:32.520
As x gets really, really,
really, really, really large,

00:02:32.520 --> 00:02:36.570
this x to the third is
going to grow much faster

00:02:36.570 --> 00:02:39.350
than this x term
right over here.

00:02:39.350 --> 00:02:41.950
So as x grows really, really,
really, really large, f of x

00:02:41.950 --> 00:02:43.866
is going to grow really,
really, really large.

00:02:43.866 --> 00:02:46.139
So the graph-- and I
don't know exactly,

00:02:46.139 --> 00:02:48.180
to see if I could plot a
couple of other points--

00:02:48.180 --> 00:02:51.890
but the bottom line is f of x
is going to approach infinity

00:02:51.890 --> 00:02:56.150
as x approaches infinity,
or f of x approaches

00:02:56.150 --> 00:03:00.660
infinity as x
approaches infinity,

00:03:00.660 --> 00:03:02.996
or as x gets larger
and larger and larger.

00:03:02.996 --> 00:03:04.620
And then what happens
as x gets smaller

00:03:04.620 --> 00:03:05.620
and smaller and smaller?

00:03:05.620 --> 00:03:08.150
If we have really
small values of x--

00:03:08.150 --> 00:03:11.960
so really negative values of
x, I should say-- once again,

00:03:11.960 --> 00:03:14.570
this, right over here,
is going to dominate.

00:03:14.570 --> 00:03:16.720
So f of x is going to
become really negative.

00:03:16.720 --> 00:03:20.520
So f of x is going to
approach negative infinity,

00:03:20.520 --> 00:03:23.800
as x approaches
negative infinity.

00:03:23.800 --> 00:03:26.690
And that is the same
behavior of g of x.

00:03:26.690 --> 00:03:30.500
As x approaches a really
large value, g of x

00:03:30.500 --> 00:03:33.340
approaches a really large value,
maybe not as fast as f of x,

00:03:33.340 --> 00:03:34.620
but it still approaches it.

00:03:34.620 --> 00:03:38.410
And likewise, as x decreases,
so does g of x decrease.

00:03:38.410 --> 00:03:41.100
It doesn't decrease as
fast as f of x might,

00:03:41.100 --> 00:03:42.860
but it's still
going to decrease.

00:03:42.860 --> 00:03:46.130
So they do seem to have the same
end behavior, at least based

00:03:46.130 --> 00:03:48.670
on the way that we
thought about it just now.

00:03:48.670 --> 00:03:52.860
Now the last option is they
have a relative maximum

00:03:52.860 --> 00:03:56.420
at the same x value.

00:03:56.420 --> 00:03:59.780
So we have to think about
what the maximum points are.

00:03:59.780 --> 00:04:02.200
Well, actually, we already
know that this is not true,

00:04:02.200 --> 00:04:04.490
because g of x has no
relative maximum points.

00:04:04.490 --> 00:04:06.067
In order to have
a maximum point,

00:04:06.067 --> 00:04:07.775
you would have to do
something like this.

00:04:07.775 --> 00:04:10.090
This, right over here,
would be a relative maximum,

00:04:10.090 --> 00:04:11.966
or, you could say, a
local maximum point.

00:04:11.966 --> 00:04:13.840
It's larger than all of
the points around it,

00:04:13.840 --> 00:04:16.510
but eventually the
function does surpass it.

00:04:16.510 --> 00:04:20.050
But this, right over
here, has no local maxima,

00:04:20.050 --> 00:04:22.760
or relative maxima,
or little bumps in it.

00:04:22.760 --> 00:04:23.940
g of x doesn't have any.

00:04:23.940 --> 00:04:26.850
So they can't have relative
maximum at the same value.

00:04:26.850 --> 00:04:30.010
So this, also, is not an option.


Comparing features of functions (example 1)

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:04.650
Which function has
no x-intercepts?

00:00:04.650 --> 00:00:06.470
So an x-intercept
is a place where

00:00:06.470 --> 00:00:09.887
the function
intersects the x-axis.

00:00:09.887 --> 00:00:11.470
And what do we know
about what's going

00:00:11.470 --> 00:00:14.000
on when something
is on the x-axis?

00:00:14.000 --> 00:00:15.930
Well, if something
is on the x-axis,

00:00:15.930 --> 00:00:19.010
then you could say
the y value is 0.

00:00:19.010 --> 00:00:21.670
Or if y is equal
to the function,

00:00:21.670 --> 00:00:24.240
you would say that the
value of the function is 0.

00:00:24.240 --> 00:00:26.750
You have an x-intercept
whenever the function

00:00:26.750 --> 00:00:28.817
itself is equal to 0.

00:00:28.817 --> 00:00:30.650
So essentially this is
equivalent to saying,

00:00:30.650 --> 00:00:33.930
which function never equals 0?

00:00:33.930 --> 00:00:38.370
So let's see if any of these
functions never equal 0.

00:00:38.370 --> 00:00:40.700
So let's look at this first
function right over here.

00:00:40.700 --> 00:00:42.199
And let me write
it right over here.

00:00:42.199 --> 00:00:48.210
So I have f of x is equal
to x squared plus 5.

00:00:48.210 --> 00:00:49.230
So this is interesting.

00:00:49.230 --> 00:00:53.770
x squared is always going
to be a non-negative number.

00:00:53.770 --> 00:00:56.060
It'll be 0 or greater.

00:00:56.060 --> 00:00:57.810
Even if x is a
negative value this

00:00:57.810 --> 00:01:00.090
is going to be 0 or greater.

00:01:00.090 --> 00:01:01.960
And 5 is obviously positive.

00:01:01.960 --> 00:01:05.200
So this whole value or
this whole expression,

00:01:05.200 --> 00:01:07.380
x squared plus 5,
is always going

00:01:07.380 --> 00:01:09.110
to be greater than
or equal to 5.

00:01:09.110 --> 00:01:11.630
So we could say f
of x is always going

00:01:11.630 --> 00:01:14.160
to be greater than
or equal to 5.

00:01:14.160 --> 00:01:17.960
So f of x is never
going to be equal to 0.

00:01:17.960 --> 00:01:20.210
If you don't believe me,
let's try it out another way.

00:01:20.210 --> 00:01:23.830
Let's set f of x equals 0
and figure out at which x

00:01:23.830 --> 00:01:25.330
that might be true.

00:01:25.330 --> 00:01:30.680
So we could say 0 is
equal to x squared plus 5.

00:01:30.680 --> 00:01:32.720
Subtract 5 from both sides.

00:01:32.720 --> 00:01:36.340
You would get negative
5 is equal to x squared.

00:01:36.340 --> 00:01:38.530
And if you take the
principal root of both sides,

00:01:38.530 --> 00:01:42.540
you get the principal root
of negative 5 is equal to x.

00:01:42.540 --> 00:01:45.040
You could even have the positive
and negative principal root

00:01:45.040 --> 00:01:46.310
of negative 5.

00:01:46.310 --> 00:01:50.630
But needless to say, if you're
dealing just with real numbers,

00:01:50.630 --> 00:01:55.350
there is no real number that is
the square root of negative 5.

00:01:55.350 --> 00:01:58.510
So f of x has no x-intercepts.

00:01:58.510 --> 00:02:01.970
So this right over here,
it meets the criteria.

00:02:01.970 --> 00:02:04.870
And this right over
here has no x-intercept.

00:02:04.870 --> 00:02:08.030
So let's see if these other
ones have x-intercepts.

00:02:08.030 --> 00:02:09.699
So remember, you
have an x-intercept

00:02:09.699 --> 00:02:13.330
if the value of the
function is 0 at some point.

00:02:13.330 --> 00:02:15.250
And we see right over
here, this function g

00:02:15.250 --> 00:02:17.410
of x is really defined
with this table.

00:02:17.410 --> 00:02:20.080
And we see it does
indeed equal 0.

00:02:20.080 --> 00:02:22.600
It happens to equal
0 when x equals 0.

00:02:22.600 --> 00:02:25.790
So it intersects the
x-axis right over there.

00:02:25.790 --> 00:02:27.670
That's its x-intercept.

00:02:27.670 --> 00:02:30.240
Now let's look at this
green function, h of x.

00:02:30.240 --> 00:02:32.270
Where does that
intersect the x-axis?

00:02:32.270 --> 00:02:34.240
Well, that's visually
more obvious.

00:02:34.240 --> 00:02:36.510
It intersects the
x-axis right over here.

00:02:36.510 --> 00:02:40.500
h of x is 0 when x is
equal to negative 6.

00:02:40.500 --> 00:02:44.000
So these last two functions
have x-intercepts.

00:02:44.000 --> 00:02:46.470
This first one does not.


Recognizing features of functions (example 1)

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:07.150
Which function increases as
x increases toward infinity

00:00:07.150 --> 00:00:13.010
and decreases as x decreases
toward negative infinity?

00:00:13.010 --> 00:00:15.430
So let's think about each
of these constraints.

00:00:15.430 --> 00:00:18.550
So, first, which function
increases as x increases?

00:00:18.550 --> 00:00:21.670
So as x increases
toward infinity.

00:00:21.670 --> 00:00:24.270
So x is going in that direction.

00:00:24.270 --> 00:00:27.270
So first let's look
at f of x over here.

00:00:27.270 --> 00:00:31.100
So f of x, as we get beyond this
minimum point right over here,

00:00:31.100 --> 00:00:36.790
as we increase our x, f of
x seems to be increasing.

00:00:36.790 --> 00:00:40.620
So f of x seems to make
the first constraint.

00:00:40.620 --> 00:00:42.610
Now let's think about g of x.

00:00:42.610 --> 00:00:45.270
Once we get past this minimum
point right over here,

00:00:45.270 --> 00:00:47.470
as x gets larger and
larger and larger,

00:00:47.470 --> 00:00:50.060
as it approaches
infinity, g of x

00:00:50.060 --> 00:00:52.790
seems to be getting larger
and larger and larger.

00:00:52.790 --> 00:00:54.340
g of x is moving up.

00:00:54.340 --> 00:00:58.410
So g of x also seems to
make this first constraint.

00:00:58.410 --> 00:00:59.980
Now let's think about h of x.

00:00:59.980 --> 00:01:03.400
As x moves towards
infinity, as x

00:01:03.400 --> 00:01:07.515
moves towards positive infinity,
h of x seems to be decreasing.

00:01:10.580 --> 00:01:13.500
So h of x does not even
make the first constraint.

00:01:13.500 --> 00:01:17.480
So our only two possibilities
are now g of x and f of x.

00:01:17.480 --> 00:01:20.650
So which of these
decrease as x decreases

00:01:20.650 --> 00:01:22.870
toward negative infinity?

00:01:22.870 --> 00:01:25.350
So let's think about
that, x decreasing

00:01:25.350 --> 00:01:26.690
toward negative infinity.

00:01:26.690 --> 00:01:29.150
We're going to be going
in that direction.

00:01:29.150 --> 00:01:31.070
So first let's look at f of x.

00:01:31.070 --> 00:01:33.360
So f of x, it kind of
goes up and down here.

00:01:33.360 --> 00:01:37.340
But after we hit this
little local maximum

00:01:37.340 --> 00:01:39.390
point-- this was a local
minimum point over here,

00:01:39.390 --> 00:01:42.240
not at a global one--
as we move to the left

00:01:42.240 --> 00:01:46.160
of this local maximum point,
as we get smaller and smaller

00:01:46.160 --> 00:01:49.970
x's, we see that the
function is decreasing.

00:01:49.970 --> 00:01:52.200
So it does seem to meet
the second constraint.

00:01:52.200 --> 00:01:56.340
It decreases as x decreases
toward negative infinity.

00:01:56.340 --> 00:01:57.980
So it meets that constraint.

00:01:57.980 --> 00:01:59.380
Now, what about g of x?

00:01:59.380 --> 00:02:01.070
After we have this
minimum point--

00:02:01.070 --> 00:02:03.700
and actually it looks like
a global minimum point--

00:02:03.700 --> 00:02:07.330
after we hit this minimum point
right over here, as x decreases

00:02:07.330 --> 00:02:10.870
toward negative infinity, g
of x seems to be increasing,

00:02:10.870 --> 00:02:12.270
not decreasing.

00:02:12.270 --> 00:02:16.330
So g of x does not meet
the second constraint.

00:02:16.330 --> 00:02:19.590
So the only function that
met both constraints seems

00:02:19.590 --> 00:02:22.069
to be f of x.


Does a vertical line represent a function?

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:05.430
In the following graph,
is y a function of x?

00:00:05.430 --> 00:00:08.320
So in order for y to
be a function of x,

00:00:08.320 --> 00:00:12.610
for any x that you input into
the function, any x for which

00:00:12.610 --> 00:00:15.370
the function is defined.

00:00:15.370 --> 00:00:17.790
So let's say we have
y is equal to f of x.

00:00:17.790 --> 00:00:19.480
So we have our little
function machine.

00:00:19.480 --> 00:00:24.190
It should spit out
exactly one value of y.

00:00:24.190 --> 00:00:27.511
If it spits out multiple values
of y, we don't know what f of x

00:00:27.511 --> 00:00:28.510
is going to be equal to.

00:00:28.510 --> 00:00:31.230
It could be equal to any of
those possible values for y.

00:00:31.230 --> 00:00:35.350
So let's see if, for this
graph, whether for a given x

00:00:35.350 --> 00:00:37.960
it spits out exactly one y.

00:00:37.960 --> 00:00:41.330
Well, the function
seems to be only defined

00:00:41.330 --> 00:00:44.360
so the domain of this function
is x is equal to negative 2.

00:00:44.360 --> 00:00:47.050
That's the only place where
we have a definition for it.

00:00:47.050 --> 00:00:50.680
And if we try to
input negative 2

00:00:50.680 --> 00:00:54.810
into this little black
box, what do we get?

00:00:54.810 --> 00:00:57.050
Do we get exactly one thing?

00:00:57.050 --> 00:00:57.550
No.

00:00:57.550 --> 00:01:00.410
If we put in negative 2
here, we could get anything.

00:01:00.410 --> 00:01:04.440
The point negative 2,
9 is on this relation.

00:01:04.440 --> 00:01:06.620
Negative 2, 8 is
on this relation.

00:01:06.620 --> 00:01:11.360
Negative 2, 7; negative 2,
7.5; negative 2, 3.14159--

00:01:11.360 --> 00:01:12.330
they're all on these.

00:01:12.330 --> 00:01:17.860
So if you put a negative 2 into
this relation, essentially,

00:01:17.860 --> 00:01:19.710
you actually get an
infinite set of values.

00:01:19.710 --> 00:01:20.540
It could be 9.

00:01:20.540 --> 00:01:22.960
It could be 3.14.

00:01:22.960 --> 00:01:24.120
It could be 8.

00:01:24.120 --> 00:01:25.840
It could be negative 8.

00:01:25.840 --> 00:01:27.990
You get an infinite
number of results.

00:01:27.990 --> 00:01:30.910
So since it does not
map to exactly one

00:01:30.910 --> 00:01:35.110
output of this function,
in the following graph,

00:01:35.110 --> 00:01:38.857
y is not a function of x.


Recognizing functions (example 3)

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:05.450
In the following table,
is y a function of x?

00:00:05.450 --> 00:00:08.170
In order for y to
be a function of x,

00:00:08.170 --> 00:00:12.920
for any x that we input into
our little function box--

00:00:12.920 --> 00:00:15.440
so let's say this is
y as a function of x.

00:00:15.440 --> 00:00:19.580
It needs to spit out
only one value of y.

00:00:19.580 --> 00:00:21.460
If it spit out
multiple values of y,

00:00:21.460 --> 00:00:23.660
then it might be a
relationship, but it's not

00:00:23.660 --> 00:00:25.060
going to be a function.

00:00:25.060 --> 00:00:26.870
So this is a function.

00:00:26.870 --> 00:00:28.350
This is a function.

00:00:28.350 --> 00:00:31.260
If we had a situation where
if we input x into a box,

00:00:31.260 --> 00:00:34.340
it could be multiple
possible y's,

00:00:34.340 --> 00:00:38.060
then this is not a function.

00:00:38.060 --> 00:00:40.820
So let's think about this
table right over here.

00:00:40.820 --> 00:00:44.690
When x is equal to 1,
we get y is equal to 1.

00:00:44.690 --> 00:00:47.270
But when x is equal to 1
again, all of a sudden,

00:00:47.270 --> 00:00:49.880
y is equal to 2.

00:00:49.880 --> 00:00:52.620
So here we have a
situation where we input 1

00:00:52.620 --> 00:00:55.520
into our little
relationship box,

00:00:55.520 --> 00:00:57.710
and when we input 1 into
our relationship box,

00:00:57.710 --> 00:01:02.270
we could get a 1, or
we could get a 2 for y.

00:01:02.270 --> 00:01:05.160
So this is definitely
not a function.

00:01:05.160 --> 00:01:06.910
For any input into
a function, it

00:01:06.910 --> 00:01:09.060
has to map to
exactly one output.

00:01:09.060 --> 00:01:11.490
Here it's mapping
to two outputs.

00:01:11.490 --> 00:01:14.650
So this is not a function.


Recognize functions from verbal descriptions (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.810 --> 00:00:03.790
Jada is ordering Mother's
Day gifts online.

00:00:03.790 --> 00:00:10.330
The shipping costs are based on
the dollar amount of the order.

00:00:10.330 --> 00:00:14.850
For orders less than
$20, shipping costs $4.

00:00:14.850 --> 00:00:18.970
For orders $20 or
more, shipping is $7.

00:00:18.970 --> 00:00:20.990
Can the dollar
amount of the order

00:00:20.990 --> 00:00:24.700
be represented as a
function of shipping costs?

00:00:24.700 --> 00:00:27.760
So they're saying,
can the dollar amount

00:00:27.760 --> 00:00:33.520
of the order-- so can
the amount of order

00:00:33.520 --> 00:00:37.870
be represented as a
function of shipping--

00:00:37.870 --> 00:00:46.130
let me do that in
that blue color costs.

00:00:46.130 --> 00:00:49.440
So if we have the shipping
costs as an input,

00:00:49.440 --> 00:00:52.400
will that map to exactly--
for a given input,

00:00:52.400 --> 00:00:55.640
will we get exactly one output
for the amount of order?

00:00:55.640 --> 00:01:01.190
In order for this to be
represented as a function,

00:01:01.190 --> 00:01:06.680
we have to input a shipping
cost, a shipping cost

00:01:06.680 --> 00:01:09.590
where this relationship
is defined.

00:01:09.590 --> 00:01:11.810
We need to input
a shipping cost,

00:01:11.810 --> 00:01:16.600
put it into our relationship,
and get exactly one dollar

00:01:16.600 --> 00:01:19.750
amount of the order in order
for this to be a function.

00:01:19.750 --> 00:01:23.020
If we get multiple dollar
amounts of the order, then

00:01:23.020 --> 00:01:25.510
the relationship, well,
it's still a relationship,

00:01:25.510 --> 00:01:27.630
but it's not going
to be a function.

00:01:27.630 --> 00:01:28.790
So let's think about it.

00:01:28.790 --> 00:01:31.390
What are the
possible inputs here?

00:01:31.390 --> 00:01:33.570
Well, there's only two
possible shipping costs.

00:01:33.570 --> 00:01:36.360
Shipping costs are
either going to be $4,

00:01:36.360 --> 00:01:39.280
or they're going to be $7.

00:01:39.280 --> 00:01:40.990
So let's think
about what happens

00:01:40.990 --> 00:01:45.120
when we input $4 in
as a shipping cost.

00:01:45.120 --> 00:01:49.190
So if we input $4
into our relationship,

00:01:49.190 --> 00:01:53.200
so we input $4 into our
little potential function box,

00:01:53.200 --> 00:01:55.400
so $4 into it,
what is the output?

00:01:55.400 --> 00:01:59.180
What is going to be the
amount of the order?

00:01:59.180 --> 00:02:05.810
Well, if the shipping cost is
$4, the amount of the order

00:02:05.810 --> 00:02:08.160
just has to be
anything less than $20.

00:02:08.160 --> 00:02:10.259
So it could have been $1.

00:02:10.259 --> 00:02:12.990
It could have been $1.50.

00:02:12.990 --> 00:02:14.990
It could've been $7.

00:02:14.990 --> 00:02:20.230
It literally can take
on any value up to $20.

00:02:20.230 --> 00:02:23.080
So it could even be $19.99.

00:02:23.080 --> 00:02:25.470
We could do a similar
thing if we input 7

00:02:25.470 --> 00:02:26.490
into this relationship.

00:02:26.490 --> 00:02:29.000
If 7 was-- and I could put
literally an infinite number

00:02:29.000 --> 00:02:29.500
of numbers.

00:02:29.500 --> 00:02:30.750
It could be a million dollars.

00:02:30.750 --> 00:02:34.960
So if I input 7 into
this relationship

00:02:34.960 --> 00:02:37.780
that we're trying to test
whether it's a function, if 7

00:02:37.780 --> 00:02:40.870
is the shipping costs,
then the order we just know

00:02:40.870 --> 00:02:43.180
is over $20, $20 or more.

00:02:43.180 --> 00:02:44.790
So it could be $20.

00:02:44.790 --> 00:02:47.190
It could be $800.

00:02:47.190 --> 00:02:49.646
It could be $1 million.

00:02:49.646 --> 00:02:51.520
There's actually an
infinite number of values

00:02:51.520 --> 00:02:53.420
that it could take
on right over here.

00:02:53.420 --> 00:02:57.480
So because for a
valid shipping cost,

00:02:57.480 --> 00:02:59.770
for each of these
valid shipping costs,

00:02:59.770 --> 00:03:02.030
I can get many, many,
many potential outputs,

00:03:02.030 --> 00:03:06.276
I don't know which output it
necessarily will output to.

00:03:06.276 --> 00:03:07.900
If someone tells you
the shipping costs

00:03:07.900 --> 00:03:12.010
and you don't necessarily
know what the order size was,

00:03:12.010 --> 00:03:13.840
this is not a function.

00:03:13.840 --> 00:03:18.170
You cannot represent the
amount of order as a function

00:03:18.170 --> 00:03:19.310
of the shipping cost.

00:03:19.310 --> 00:03:21.420
So, no.


Recognize functions from verbal descriptions (example 1)

WEBVTT
Kind: captions
Language: en

00:00:00.840 --> 00:00:06.370
The value of y is always
3 more than twice x.

00:00:06.370 --> 00:00:15.470
So we can say that y is
equal to 3 more than twice x.

00:00:15.470 --> 00:00:19.120
So it's 3 plus 2x is another way
of saying this first sentence.

00:00:19.120 --> 00:00:22.320
So is y a function of x?

00:00:22.320 --> 00:00:24.090
So whenever you're
asked whether something

00:00:24.090 --> 00:00:27.090
is a function of something
else, you're really just saying,

00:00:27.090 --> 00:00:32.880
look, for any input x, does
it map to exactly one y?

00:00:32.880 --> 00:00:36.320
So if we say y is a function
of x, in order for this to be

00:00:36.320 --> 00:00:40.080
a function for any x that
you input into this function,

00:00:40.080 --> 00:00:44.670
you must get exactly one y.

00:00:44.670 --> 00:00:49.490
So if you input an x you
must get exactly one y value.

00:00:49.490 --> 00:00:52.170
If you got two values, then
it's no longer a function.

00:00:52.170 --> 00:00:55.460
For any input, you
get exactly one y.

00:00:55.460 --> 00:00:58.360
You could have two inputs
that get to the same y,

00:00:58.360 --> 00:01:00.390
but you can't have
one input that

00:01:00.390 --> 00:01:02.070
results in two
different outputs.

00:01:02.070 --> 00:01:05.800
You don't know what the function
is valued at at that input.

00:01:05.800 --> 00:01:08.540
Now, here it looks pretty
clear that for any input,

00:01:08.540 --> 00:01:10.850
you get exactly one output.

00:01:10.850 --> 00:01:13.470
Any input uniquely
determines which y.

00:01:13.470 --> 00:01:15.090
It's not like if you
put an x in here,

00:01:15.090 --> 00:01:16.929
you're not sure what
y is going to be.

00:01:16.929 --> 00:01:18.220
You know what y is going to be.

00:01:18.220 --> 00:01:19.970
If x is 0, y is 3.

00:01:19.970 --> 00:01:23.090
If x is 1, y is 5.

00:01:23.090 --> 00:01:26.230
And so this is definitely
a function of x. y

00:01:26.230 --> 00:01:29.593
is definitely a function of x.


Recognizing functions (example 4)

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:05.670
In the relation x is
equal to y squared plus 3,

00:00:05.670 --> 00:00:10.640
can y be represented as a
mathematical function of x?

00:00:10.640 --> 00:00:12.440
So the way they've
written it, x is

00:00:12.440 --> 00:00:15.680
being represented as a
mathematical function of y.

00:00:15.680 --> 00:00:18.780
We could even say that
x as a function of y

00:00:18.780 --> 00:00:21.697
is equal to y squared plus 3.

00:00:21.697 --> 00:00:23.530
Now, let's see if we
can do it the other way

00:00:23.530 --> 00:00:26.351
around, if we can represent
y as a function of x.

00:00:26.351 --> 00:00:28.100
So one way you could
think about it is you

00:00:28.100 --> 00:00:30.490
could essentially try
to solve for y here.

00:00:30.490 --> 00:00:32.060
So let's do that.

00:00:32.060 --> 00:00:36.720
So I have x is equal
to y squared plus 3.

00:00:36.720 --> 00:00:39.810
Subtract 3 from both
sides, you get x minus 3

00:00:39.810 --> 00:00:42.710
is equal to y squared.

00:00:42.710 --> 00:00:46.260
Now, the next step is going
to be tricky, x minus 3

00:00:46.260 --> 00:00:49.000
is equal to y squared.

00:00:49.000 --> 00:00:53.110
So y could be equal to-- and I'm
just going to swap the sides.

00:00:53.110 --> 00:00:56.270
y could be equal to-- if we take
the square root of both sides,

00:00:56.270 --> 00:00:59.850
it could be the positive
square root of x minus 3,

00:00:59.850 --> 00:01:02.700
or it could be the
negative square root.

00:01:02.700 --> 00:01:07.760
Or y could be the negative
square root of x minus 3.

00:01:07.760 --> 00:01:10.270
If you don't believe me,
square both sides of this.

00:01:10.270 --> 00:01:12.890
You'll get y squared
is equal to x minus 3.

00:01:12.890 --> 00:01:14.470
Square both sides
of this, you're

00:01:14.470 --> 00:01:16.750
going to get y squared
is equal to-- well,

00:01:16.750 --> 00:01:19.270
the negative squared is just
going to be a positive 1.

00:01:19.270 --> 00:01:23.750
And you're going to get y
squared is equal to x minus 3.

00:01:23.750 --> 00:01:27.750
So this is a situation
here where for a given x,

00:01:27.750 --> 00:01:30.520
you could actually
have 2 y-values.

00:01:30.520 --> 00:01:32.960
Let me show you.

00:01:32.960 --> 00:01:35.700
Let me attempt to
sketch this graph.

00:01:35.700 --> 00:01:38.520
So let's say this is our y-axis.

00:01:38.520 --> 00:01:40.850
I guess I could call
it this relation.

00:01:40.850 --> 00:01:43.660
This is our x-axis.

00:01:43.660 --> 00:01:46.480
And this right over here,
y is a positive square root

00:01:46.480 --> 00:01:47.849
of x minus 3.

00:01:47.849 --> 00:01:49.140
That's going to look like this.

00:01:49.140 --> 00:01:53.040
So if this is x is equal to 3,
it's going to look like this.

00:01:53.040 --> 00:01:56.100
That's y is equal to the
positive square root of x

00:01:56.100 --> 00:01:57.530
minus 3.

00:01:57.530 --> 00:02:00.220
And this over here, y is equal
to the negative square root

00:02:00.220 --> 00:02:02.390
of x minus 3, is going to
look something like this.

00:02:04.962 --> 00:02:07.170
I should make it a little
bit more symmetric looking,

00:02:07.170 --> 00:02:09.128
because it's going to
essentially be the mirror

00:02:09.128 --> 00:02:11.090
image if you flip
over the x-axis.

00:02:11.090 --> 00:02:14.390
So it's going to look
something like this-- y is

00:02:14.390 --> 00:02:18.330
equal to the negative
square root of x minus 3.

00:02:18.330 --> 00:02:22.220
And this right over here,
this relationship cannot be--

00:02:22.220 --> 00:02:26.070
this right over here
is not a function of x.

00:02:26.070 --> 00:02:28.850
In order to be a function
of x, for a given x

00:02:28.850 --> 00:02:32.840
it has to map to exactly
one value for the function.

00:02:32.840 --> 00:02:36.410
But here you see it's mapping
to two values of the function.

00:02:36.410 --> 00:02:40.810
So, for example, let's say
we take x is equal to 4.

00:02:40.810 --> 00:02:46.230
So x equals 4 could get
us to y is equal to 1.

00:02:46.230 --> 00:02:48.250
4 minus 3 is 1.

00:02:48.250 --> 00:02:51.010
Take the positive square
root, it could be 1.

00:02:51.010 --> 00:02:56.270
Or you could have x equals 4,
and y is equal to negative 1.

00:02:56.270 --> 00:02:58.410
So you can't have
this situation.

00:02:58.410 --> 00:03:04.900
If you were making a table
x and y as a function of x,

00:03:04.900 --> 00:03:08.180
you can't have x is equal to 4.

00:03:08.180 --> 00:03:10.240
And at one point it equals 1.

00:03:10.240 --> 00:03:13.050
And then in another
interpretation of it,

00:03:13.050 --> 00:03:15.830
when x is equal to 4,
you get to negative 1.

00:03:15.830 --> 00:03:19.980
You can't have one input
mapping to two outputs

00:03:19.980 --> 00:03:22.120
and still be a function.

00:03:22.120 --> 00:03:26.130
So in this case, the relation
cannot-- for this relation,

00:03:26.130 --> 00:03:31.730
y cannot be represented as a
mathematical function of x.


Recognizing features of functions (example 3)

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:05.345
Which function has a local
minimum with a value of 1?

00:00:09.635 --> 00:00:11.260
So let's think about
this a little bit.

00:00:11.260 --> 00:00:13.040
So they tell us the
x values, and we

00:00:13.040 --> 00:00:16.820
see x increasing from negative
4 all the way to positive 4.

00:00:16.820 --> 00:00:20.550
So our x values are increasing.

00:00:20.550 --> 00:00:24.100
And they give us
different functions here.

00:00:24.100 --> 00:00:26.350
And they essentially say
which of these functions

00:00:26.350 --> 00:00:31.910
have a local minimum value where
the value of the minimum is 1?

00:00:31.910 --> 00:00:35.190
So they hit a
minimum point at 1.

00:00:35.190 --> 00:00:37.100
So that means that the
value of the function

00:00:37.100 --> 00:00:39.250
must be 1 at that minimum point.

00:00:39.250 --> 00:00:41.540
So we can see candidate
points right over here.

00:00:41.540 --> 00:00:44.590
F of x hits 1 right over there.

00:00:44.590 --> 00:00:47.760
G of x hits 1 right over there.

00:00:47.760 --> 00:00:51.530
And h of x hits 1
right over there.

00:00:51.530 --> 00:00:53.790
And let's see, that's the
only time that they hit 1.

00:00:53.790 --> 00:00:57.220
Now let's think about which
of these are minimum values.

00:00:57.220 --> 00:01:00.800
So on f of x, when we go from
negative 2 to negative 1,

00:01:00.800 --> 00:01:05.530
we are increasing from 0 to
1 and then we increase again.

00:01:05.530 --> 00:01:07.620
Well, that's not going
to be a minimum value.

00:01:07.620 --> 00:01:08.670
We are increasing.

00:01:08.670 --> 00:01:13.830
So let me draw a little
axis right over here.

00:01:13.830 --> 00:01:17.530
So let me actually draw
the whole thing just

00:01:17.530 --> 00:01:19.990
to make it clear
what we're doing.

00:01:19.990 --> 00:01:22.940
So we have negative 4, 3, 2, 1.

00:01:22.940 --> 00:01:25.160
Positive 1, 2, 3, 4.

00:01:25.160 --> 00:01:27.400
And of course this is 0,
negative 4, negative 3,

00:01:27.400 --> 00:01:29.120
negative 2, negative 1.

00:01:29.120 --> 00:01:33.891
1, 2, 3, 4, and this is going
to be 0 right over here.

00:01:33.891 --> 00:01:35.390
And we're going to
care about what's

00:01:35.390 --> 00:01:39.550
happening when these
functions hit 1.

00:01:39.550 --> 00:01:44.710
So f of x, when x is
equal to negative 2, is 0.

00:01:44.710 --> 00:01:47.200
When x is equal to
negative 1 is 1,

00:01:47.200 --> 00:01:51.340
and then when x is
equal to 0 is 5.

00:01:51.340 --> 00:01:54.831
So 2, 3, 4, 5 is that
right over there.

00:01:54.831 --> 00:01:57.330
And if we assume that these are
sampling from the function--

00:01:57.330 --> 00:01:59.880
we actually necessarily can't
make that assumption here--

00:01:59.880 --> 00:02:02.020
but if we assume that
these are sampling, then

00:02:02.020 --> 00:02:03.510
if this was a
continuous function,

00:02:03.510 --> 00:02:05.301
then we would assume
that the function does

00:02:05.301 --> 00:02:06.210
something like this.

00:02:06.210 --> 00:02:08.590
And so it's clearly not a
minimum point right here.

00:02:08.590 --> 00:02:15.520
We're increasing into 1.

00:02:15.520 --> 00:02:17.860
So this is not a minimum point.

00:02:17.860 --> 00:02:19.720
Now let's look at g of x.

00:02:19.720 --> 00:02:22.075
Let's see, when g of
x is 1, we're at 2.

00:02:24.580 --> 00:02:27.600
When g of x is at 2, we're at 1.

00:02:27.600 --> 00:02:30.480
And when g of x is 3,
we're at negative 1.

00:02:30.480 --> 00:02:31.950
So we are at negative 1.

00:02:31.950 --> 00:02:33.320
So this is the opposite.

00:02:33.320 --> 00:02:37.930
We do decrease into that value
when the function hits 1,

00:02:37.930 --> 00:02:39.114
but then we keep decreasing.

00:02:39.114 --> 00:02:40.780
In order for this to
be a minimum point,

00:02:40.780 --> 00:02:45.740
we would have to decrease into
it and then increase out of it.

00:02:45.740 --> 00:02:48.590
This right over here is
also not a minimum point.

00:02:48.590 --> 00:02:50.990
Now let's look at h of x.

00:02:50.990 --> 00:02:55.400
When x is negative
4, we're at 5.

00:02:55.400 --> 00:02:58.960
When x is at negative
3, we are at 1.

00:02:58.960 --> 00:03:03.390
And when x is at negative
2, we jump back up to 3.

00:03:03.390 --> 00:03:05.370
So this looks like
a minimum point.

00:03:05.370 --> 00:03:07.360
We decreased going
into it and then

00:03:07.360 --> 00:03:08.810
we increased going out of it.

00:03:08.810 --> 00:03:10.919
So this is our
local minimum point.

00:03:10.919 --> 00:03:12.460
And if this was a
continuous function

00:03:12.460 --> 00:03:17.740
it maybe might look
something like that.

00:03:17.740 --> 00:03:21.001
So we will go with h of x.


Recognizing features of functions (example 2)

WEBVTT
Kind: captions
Language: en

00:00:00.730 --> 00:00:03.900
Which of these functions is odd?

00:00:03.900 --> 00:00:05.750
And so let's remind
ourselves what

00:00:05.750 --> 00:00:07.841
it means for a
function to be odd.

00:00:07.841 --> 00:00:09.340
So I have a function--
well, they've

00:00:09.340 --> 00:00:12.422
already used f, g,
and h, so I'll use j.

00:00:12.422 --> 00:00:14.660
So function j is odd.

00:00:14.660 --> 00:00:19.280
If you evaluate j at some
value-- so let's say j of a.

00:00:19.280 --> 00:00:23.760
And if you evaluate that j at
the negative of that value,

00:00:23.760 --> 00:00:27.170
and if these two things are
the negative of each other,

00:00:27.170 --> 00:00:29.400
then my function is odd.

00:00:29.400 --> 00:00:31.100
If these two things
were the same--

00:00:31.100 --> 00:00:32.974
if they didn't have this
negative here-- then

00:00:32.974 --> 00:00:34.770
it would be an even function.

00:00:34.770 --> 00:00:38.930
So let's see which of these
meet the criteria of being odd.

00:00:38.930 --> 00:00:41.150
So let's look at f of x.

00:00:41.150 --> 00:00:43.190
So we could pick a
particular point.

00:00:43.190 --> 00:00:44.810
So let's say when
x is equal to 2.

00:00:44.810 --> 00:00:49.750
So we get f of 2 is equal to 2.

00:00:49.750 --> 00:00:51.670
Now, what is f of negative 2?

00:00:51.670 --> 00:00:55.160
f of negative 2
looks like it is 6.

00:00:55.160 --> 00:00:58.180
f of negative 2 is equal to 6.

00:00:58.180 --> 00:01:00.210
So these aren't the
negative of each other.

00:01:00.210 --> 00:01:03.275
In order for this to
be odd, f of negative 2

00:01:03.275 --> 00:01:05.650
would have had to be equal to
the negative of this, would

00:01:05.650 --> 00:01:08.020
have had to be
equal to negative 2.

00:01:08.020 --> 00:01:13.300
So f of x is definitely not odd.

00:01:13.300 --> 00:01:16.410
So all I have to do is
find even one case that

00:01:16.410 --> 00:01:18.820
violated this
constraint to be odd.

00:01:18.820 --> 00:01:22.110
And so I can say it's
definitely not odd.

00:01:22.110 --> 00:01:26.460
Now let's look at g of x.

00:01:26.460 --> 00:01:30.360
So I could use the same-- let's
see, when x is equal to 2,

00:01:30.360 --> 00:01:34.400
we get g of 2 is
equal to negative 7.

00:01:34.400 --> 00:01:37.460
Now let's look at
when g is negative 2.

00:01:37.460 --> 00:01:43.630
So we get g of negative 2
is also equal to negative 7.

00:01:43.630 --> 00:01:45.680
So here we have a
situation-- and it

00:01:45.680 --> 00:01:49.020
looks like that's the case for
any x we pick-- that g of x

00:01:49.020 --> 00:01:51.990
is going to be equal
to g of negative x.

00:01:51.990 --> 00:01:56.400
So g of x is equal
to g of negative x.

00:01:56.400 --> 00:01:59.272
It's symmetric
around the y-- or I

00:01:59.272 --> 00:02:01.230
should say the vertical
axis-- right over here.

00:02:01.230 --> 00:02:04.172
So g of x is even, not odd.

00:02:04.172 --> 00:02:05.630
So which of these
functions is odd?

00:02:05.630 --> 00:02:07.770
Definitely not g of x.

00:02:07.770 --> 00:02:09.729
So our last hope is h of x.

00:02:09.729 --> 00:02:14.560
Let's see if h of x seems
to meet the criteria.

00:02:14.560 --> 00:02:16.410
I'll do it in this green color.

00:02:16.410 --> 00:02:21.730
So if we take h of 1-- and we
can look at it even visually.

00:02:21.730 --> 00:02:25.660
So h of 1 gets us right
over here. h of negative 1

00:02:25.660 --> 00:02:30.060
seems to get us an equal amount,
an equal distance, negative.

00:02:30.060 --> 00:02:32.330
So it seems to fit for 1.

00:02:32.330 --> 00:02:35.780
For 2-- well, 2
is at the x-axis.

00:02:35.780 --> 00:02:39.220
But that's definitely h of 2
is 0. h of negative 2 is 0.

00:02:39.220 --> 00:02:40.970
But those are the
negatives of each other.

00:02:40.970 --> 00:02:42.940
0 is equal to negative 0.

00:02:42.940 --> 00:02:48.210
If we go to, say, h of 4, h
of 4 is this negative number.

00:02:48.210 --> 00:02:53.680
And h of negative 4 seems
to be a positive number

00:02:53.680 --> 00:02:55.120
of the same magnitude.

00:02:55.120 --> 00:02:59.420
So once again, this is
the negative of this.

00:02:59.420 --> 00:03:03.840
So it looks like this is
indeed an odd function.

00:03:03.840 --> 00:03:07.340
And another way to visually
spot an odd function

00:03:07.340 --> 00:03:10.490
is a function-- it's going
to go through the origin,

00:03:10.490 --> 00:03:14.080
and you could essentially
flip it over on both axes.

00:03:14.080 --> 00:03:17.700
So if you flip this, the right
half, over the left half,

00:03:17.700 --> 00:03:20.660
and then flip that over
the horizontal axis,

00:03:20.660 --> 00:03:22.700
you are going to get
this right over here.

00:03:22.700 --> 00:03:24.847
So you see here we're
going up and to the right.

00:03:24.847 --> 00:03:26.680
Here we're going to go
down and to the left.

00:03:26.680 --> 00:03:28.180
And then you curve
right over there.

00:03:28.180 --> 00:03:29.685
You curve up just like that.

00:03:29.685 --> 00:03:32.060
But the easiest way to test
it is just to do what we did,

00:03:32.060 --> 00:03:33.550
look at a given x.

00:03:33.550 --> 00:03:37.020
So for example, when x
is equal to 8, h of 8

00:03:37.020 --> 00:03:40.050
looks like this
number right around 8.

00:03:40.050 --> 00:03:44.520
h of negative 8 looks like it's
pretty close to negative 8.

00:03:44.520 --> 00:03:46.590
So they seem to be the
negative of each other.

00:03:46.590 --> 00:03:48.882
It sounds like a car crash
just happened outside.

00:03:48.882 --> 00:03:50.340
Anyway, hopefully
you enjoyed that.

00:03:50.340 --> 00:03:52.910
Not the car crash,
the math problem.


Interpreting features of functions (example 2)

WEBVTT
Kind: captions
Language: en

00:00:01.120 --> 00:00:02.770
Cid is experimenting
with a piece

00:00:02.770 --> 00:00:04.650
of sandpaper and some wood.

00:00:04.650 --> 00:00:06.870
He tries scraping the
piece of sandpaper

00:00:06.870 --> 00:00:10.530
over the wood in different ways
to see how much is scraped off.

00:00:10.530 --> 00:00:13.390
The thickness of wood
scraped off, in millimeters,

00:00:13.390 --> 00:00:16.800
as a function of the
speed of the sandpaper,

00:00:16.800 --> 00:00:18.440
in meters per second.

00:00:18.440 --> 00:00:19.610
t of v.

00:00:19.610 --> 00:00:23.230
So this is the
thickness scraped off.

00:00:23.230 --> 00:00:26.650
So that's the thickness,
or how much is scraped off.

00:00:26.650 --> 00:00:29.990
And it is a function of speed.

00:00:29.990 --> 00:00:32.910
One, they're using v and also
they're getting negative value.

00:00:32.910 --> 00:00:34.840
So we care about the direction.

00:00:34.840 --> 00:00:36.730
It's actually the velocity.

00:00:36.730 --> 00:00:39.900
So this is how much is scraped
off as a function of velocity.

00:00:39.900 --> 00:00:41.350
It's shown below.

00:00:41.350 --> 00:00:44.230
And so if the velocity
is greater than 0,

00:00:44.230 --> 00:00:47.420
that means that the sandpaper
is moving to the right.

00:00:47.420 --> 00:00:48.160
That makes sense.

00:00:48.160 --> 00:00:49.730
That's the standard convention.

00:00:49.730 --> 00:00:51.335
And if the velocity
is less than 0,

00:00:51.335 --> 00:00:54.080
it means the sandpaper
is moving to the left.

00:00:54.080 --> 00:00:55.170
Fair enough.

00:00:55.170 --> 00:00:56.860
The function is even.

00:00:56.860 --> 00:01:01.050
What is the significance of
the evenness of this function?

00:01:01.050 --> 00:01:05.330
Well, the fact that it's
even means that t of v

00:01:05.330 --> 00:01:10.220
is equal to t of negative
v. So that tells us

00:01:10.220 --> 00:01:18.640
that if our velocity is 8
meters per second to the left

00:01:18.640 --> 00:01:20.640
we're going to get
as much scraped off

00:01:20.640 --> 00:01:23.937
as if we go 8 millimeters
per second to the right.

00:01:23.937 --> 00:01:25.270
And we see that right over here.

00:01:25.270 --> 00:01:28.430
So that is equal to that.

00:01:28.430 --> 00:01:31.550
If we go at 6 meters
per second to the left

00:01:31.550 --> 00:01:33.660
we're going to get just
as much scraped off

00:01:33.660 --> 00:01:36.430
as we go 6 millimeters--
6 meters per second,

00:01:36.430 --> 00:01:39.020
these are in meters per
second-- to the right.

00:01:39.020 --> 00:01:41.490
So these two are
going to be the same.

00:01:41.490 --> 00:01:43.820
So it's really telling
us-- and we could say do it

00:01:43.820 --> 00:01:46.360
for 4 meters per
second and negative 4--

00:01:46.360 --> 00:01:49.150
is it doesn't matter if we
go to the left or the right.

00:01:49.150 --> 00:01:52.520
What really matters is the
magnitude of the velocity

00:01:52.520 --> 00:01:54.407
or the absolute value of it.

00:01:54.407 --> 00:01:56.990
But it doesn't matter if we're
going to the left or the right.

00:01:56.990 --> 00:02:00.700
Whether we're going to
the left or the right

00:02:00.700 --> 00:02:03.930
for a given
magnitude of velocity

00:02:03.930 --> 00:02:06.150
we are going to get the
same amount scraped off.

00:02:06.150 --> 00:02:07.650
Now let's see which
of these choices

00:02:07.650 --> 00:02:10.009
are consistent with
what I just said.

00:02:10.009 --> 00:02:13.300
Moving the sandpaper faster
scrapes off more wood.

00:02:13.300 --> 00:02:14.530
Well, that's true.

00:02:14.530 --> 00:02:18.120
We see as the speed increases,
or the magnitude of the speed

00:02:18.120 --> 00:02:20.490
increases, we scrape
off more wood.

00:02:20.490 --> 00:02:22.919
As the magnitude of the
speed, this negative 8,

00:02:22.919 --> 00:02:24.960
you might say, hey, that's
lower than negative 2,

00:02:24.960 --> 00:02:27.110
but the magnitude is larger.

00:02:27.110 --> 00:02:29.570
We're going 8 meters
per second to the left

00:02:29.570 --> 00:02:31.040
and we're scraping off more.

00:02:31.040 --> 00:02:34.470
So this is a true
statement, but it's not

00:02:34.470 --> 00:02:38.590
the significance of the
evenness of the function.

00:02:38.590 --> 00:02:41.630
This could have been true
even if this was a seven,

00:02:41.630 --> 00:02:44.690
but then this function
would no longer be even.

00:02:44.690 --> 00:02:48.290
The piece of wood is
six millimeters thick.

00:02:48.290 --> 00:02:50.900
We actually don't get any
of that from the function.

00:02:50.900 --> 00:02:52.440
Moving the sandpaper
to the right

00:02:52.440 --> 00:02:54.750
has the same effect as
moving it to the left.

00:02:54.750 --> 00:02:57.470
Well, that seems pretty close
to what I had said earlier.

00:02:57.470 --> 00:03:00.610
That for a given speed to
the right or to the left,

00:03:00.610 --> 00:03:03.090
we get the same amount
that is taken off

00:03:03.090 --> 00:03:05.910
of the piece of sandpaper,
or the piece of wood.

00:03:05.910 --> 00:03:08.300
So this looks like our answer.

00:03:08.300 --> 00:03:11.320
Keeping the sandpaper still
doesn't scrape off any wood.

00:03:11.320 --> 00:03:13.420
Well, that is true
but once again

00:03:13.420 --> 00:03:17.780
is not the significance of
the evenness of this function.


Interpreting the y-intercept of a graph in context

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:04.950
Mr. Theisen is honing his
deadly three-point precision

00:00:04.950 --> 00:00:06.670
on the basketball court.

00:00:06.670 --> 00:00:10.190
For one of his shots,
the height of the ball

00:00:10.190 --> 00:00:18.150
in feet as a function of
horizontal distance, in feet,

00:00:18.150 --> 00:00:21.760
y of x-- so here y
is a function of x.

00:00:21.760 --> 00:00:24.930
So the height must
be y because that's

00:00:24.930 --> 00:00:27.460
the thing that is a
function of something else.

00:00:27.460 --> 00:00:29.930
So this right over
here is height.

00:00:29.930 --> 00:00:34.070
So our y-axis is going
to represent height.

00:00:34.070 --> 00:00:35.890
And it is a function of x.

00:00:35.890 --> 00:00:38.620
So x must represent
horizontal distance

00:00:38.620 --> 00:00:41.780
because height is a function
of horizontal distance.

00:00:41.780 --> 00:00:46.520
So this right over here
is horizontal distance.

00:00:51.900 --> 00:00:53.610
Now, it's plotted below.

00:00:53.610 --> 00:00:57.180
Mr Theisen is standing
at x equals 0.

00:00:57.180 --> 00:00:58.790
So he's standing
right over here.

00:00:58.790 --> 00:01:01.850
This is Mr. Theisen, as I
draw my best attempt to draw

00:01:01.850 --> 00:01:06.370
a little stick figure
version of Mr. Theisen.

00:01:06.370 --> 00:01:09.180
That's not even an acceptable
stick figure right over there.

00:01:09.180 --> 00:01:14.110
So this is Mr. Theisen, and
he's standing at x equals 0.

00:01:14.110 --> 00:01:16.990
And at x equals 0, he is
shooting a basketball.

00:01:20.550 --> 00:01:23.470
And you see from the
function right over here

00:01:23.470 --> 00:01:28.680
that where the graph
intersects the y-axis,

00:01:28.680 --> 00:01:32.340
that tells us that's essentially
the height of the ball

00:01:32.340 --> 00:01:35.814
when x is equal 0, where it's
where Mr. Theisen is standing.

00:01:35.814 --> 00:01:37.230
And if we look at
this, this looks

00:01:37.230 --> 00:01:41.540
like it's 2, 4, 6 feet high.

00:01:41.540 --> 00:01:43.690
So that's really the
initial position of the ball

00:01:43.690 --> 00:01:46.420
when Mr. Theisen is
about to let go of it.

00:01:46.420 --> 00:01:48.020
Then he lets go of
it, and the ball

00:01:48.020 --> 00:01:50.960
goes in this
parabolic trajectory.

00:01:50.960 --> 00:01:53.590
It increases, increases,
increasing, increases.

00:01:53.590 --> 00:01:57.920
It looks like it hits a maximum
point right around there,

00:01:57.920 --> 00:01:58.720
roughly.

00:01:58.720 --> 00:02:01.560
That looks like it's
at about 16 feet.

00:02:01.560 --> 00:02:03.420
And then it starts to go down.

00:02:03.420 --> 00:02:04.970
And right over
here-- and this looks

00:02:04.970 --> 00:02:06.910
like it's about,
let's see, 22, 24,

00:02:06.910 --> 00:02:10.160
26 feet out-- it looks
like it hit something.

00:02:10.160 --> 00:02:12.432
And considering that
something is 10 feet high,

00:02:12.432 --> 00:02:14.640
it's reasonable to assume
that the thing that it hits

00:02:14.640 --> 00:02:15.650
is the goal.

00:02:15.650 --> 00:02:18.940
And especially
because the question

00:02:18.940 --> 00:02:21.990
states that he has deadly
three-point precision,

00:02:21.990 --> 00:02:24.110
we can assume it's not
crazy that he actually

00:02:24.110 --> 00:02:24.822
makes the goal.

00:02:24.822 --> 00:02:26.530
And so that's where
it goes into the net.

00:02:26.530 --> 00:02:28.400
And then the net forces
the ball to go down

00:02:28.400 --> 00:02:32.480
at a much steeper trajectory.

00:02:32.480 --> 00:02:34.600
And this is exactly, of
course, 10 feet high,

00:02:34.600 --> 00:02:36.240
the height of the goal.

00:02:36.240 --> 00:02:38.817
Now let's see which of
these interpretations

00:02:38.817 --> 00:02:40.400
are consistent with
the interpretation

00:02:40.400 --> 00:02:42.060
that we just did.

00:02:42.060 --> 00:02:44.400
The ball is released
from Mr. Theisen's hand

00:02:44.400 --> 00:02:46.030
at a height of 6 feet.

00:02:46.030 --> 00:02:47.380
Well, that looks exactly right.

00:02:47.380 --> 00:02:50.571
When x is equal to 0,
the ball is 6 feet.

00:02:50.571 --> 00:02:52.070
And not only is
that right, but that

00:02:52.070 --> 00:02:55.290
is the significance of the
y-intercept of this function.

00:02:55.290 --> 00:02:58.050
The y-intercept
is the value of y,

00:02:58.050 --> 00:03:01.600
the height when x is equal to 0.

00:03:01.600 --> 00:03:05.140
So that is indeed
the significance

00:03:05.140 --> 00:03:06.360
of the y-intercept.

00:03:06.360 --> 00:03:07.890
Let's look at
these other things.

00:03:07.890 --> 00:03:11.550
Mr. Theisen is shooting the
basketball from 26 feet away.

00:03:11.550 --> 00:03:13.100
Well, that's right.

00:03:13.100 --> 00:03:14.380
He's at x equals 0.

00:03:14.380 --> 00:03:16.550
The goal is at 26 feet away.

00:03:16.550 --> 00:03:19.836
But that's not the significance
of the y-intercept.

00:03:19.836 --> 00:03:21.460
That would be the
significance of where

00:03:21.460 --> 00:03:25.210
we saw this little point here
where the ball dropped down

00:03:25.210 --> 00:03:27.040
at a steeper angle.

00:03:27.040 --> 00:03:29.570
The rim of the basketball
hoop is 10 feet high.

00:03:29.570 --> 00:03:31.289
Once again, that's true.

00:03:31.289 --> 00:03:32.080
You can look at it.

00:03:32.080 --> 00:03:33.413
You can see it right over there.

00:03:33.413 --> 00:03:36.100
But that's not the significance
of the y-intercept.

00:03:36.100 --> 00:03:38.350
The maximum height that the
ball reaches is 16 feet.

00:03:38.350 --> 00:03:41.470
Well, once again, that
is true, but that's

00:03:41.470 --> 00:03:43.830
the significance of this
maximum point on the curve.

00:03:43.830 --> 00:03:46.650
That's not the significance
of the y-intercept.

00:03:46.650 --> 00:03:49.480
So we'll go with
this first choice.