- THE EQUATION Y = -ONE 
SIXTEENTH X SQUARED + 4X +3
MODELS THE HEIGHT OF AN ARROW
WHERE X IS THE HORIZONTAL 
DISTANCE IN FEET
FROM THE POINT 
THE ARROW IS SHOT.
SO Y REPRESENTS THE HEIGHT 
IN FEET
AND X REPRESENTS 
THE HORIZONTAL DISTANCE
TRAVELED IN FEET.
WE WANT TO ANSWER 
THE FOLLOWING THREE QUESTIONS.
NUMBER ONE, HOW HIGH 
IS THE ARROW WHEN IT IS SHOT,
NUMBER TWO, WHAT IS THE 
MAXIMUM HEIGHT OF THE ARROW,
AND NUMBER THREE,
HOW FAR HORIZONTALLY 
DOES THE ARROW TRAVEL
BEFORE HITTING THE GROUND.
SO LET'S GO THROUGH THESE ONE 
AT A TIME.
TO ANSWER THE FIRST QUESTION
OF HOW HIGH THE ARROW IS 
WHEN IT IS SHOT,
RIGHT AT THE INSTANT 
THE ARROW IS SHOT,
THE ARROW HAS NOT TRAVELED 
HORIZONTALLY
THEREFORE, TO ANSWER 
THIS QUESTION,
WE'RE GOING TO SET X EQUAL 
TO 0 AND THEN SOLVE FOR Y.
SO IF WE LET X = 0, 
WE WANT TO EVALUATE Y OF 0
WHICH WOULD BE -ONE SIXTEENTH 
x 0 SQUARED + 4X 0 +3
SO Y OF 0 IS JUST GOING TO BE 
0 + 0 +3 OR 3.
THEREFORE THE ARROW STARTS 
AT A HEIGHT OF THREE FEET.
NUMBER TWO,
WE WANT TO DETERMINE THE 
MAXIMUM HEIGHT OF THE ARROW.
NOTICE OUR EQUATION 
IS A QUADRATIC EQUATION
AND THEREFORE THE GRAPH 
IS A PARABOLA
AND BECAUSE "A" IS NEGATIVE,
IT'S A PARABOLA 
THAT OPENS DOWN
OR LOOKS SOMETHING LIKE THIS.
SO THE MAXIMUM HEIGHT OF 
THE ARROW WOULD BE REPRESENTED
BY THE VERTEX.
SO WELL FIND THE VERTEX
WHERE THE X COORDINATE 
WILL REPRESENT
THE HORIZONTAL DISTANCE 
TRAVELED
AND THE Y COORDINATE 
WILL BE THE MAXIMUM HEIGHT.
REMEMBER THE COORDINATES 
OF THE VERTEX
ARE -B OVER 2A, 
F OF -B OVER 2A.
SO WE'LL START BY DETERMINING 
THE X COORDINATE OF THE VERTEX
USING THIS FORMULA HERE.
NOTICE "A" 
IS EQUAL TO -ONE SIXTEENTH,
B IS EQUAL TO POSITIVE 4, 
AND WE DON'T NEED C,
BUT C IS EQUAL TO 3.
SO THE X COORDINATE IS GOING 
TO BE EQUAL TO -B OR -4
DIVIDED BY 2 TIMES "A" 
OR 2 TIMES -ONE SIXTEENTH.
SO LET'S PUT THIS 
TWO OVER ONE,
WE'RE GOING TO HAVE -4 DIVIDED 
BY THE 2 AND THE 16 SIMPLIFY,
THIS SIMPLIFIES THE 1, 
THIS SIMPLIFIES THE 8.
SO WE HAVE -ONE EIGHTH.
SO WE'LL HAVE -4 x 
THE RECIPROCAL OF -ONE EIGHTH
WHICH IS JUST -8, 
OR -8 OVER 1.
SO X IS EQUAL TO 32 
OR 32 FEET.
AGAIN THIS IS THE HORIZONTAL 
DISTANCE TRAVELED
AT THE MAXIMUM HEIGHT.
SO TO FIND THE MAXIMUM HEIGHT,
WE'LL NOW HAVE TO SUBSTITUTE 
X = 32
BACK INTO THE ORIGINAL 
EQUATION.
SO WE WANT TO FIND Y OF 32,
WHICH IS GOING TO BE EQUAL TO 
-ONE SIXTEENTH x 32 SQUARED,
+ 4 x 32 +3.
LET'S EVALUATE THIS 
ON THE CALCULATOR.
SO WE HAVE -ONE SIXTEENTH, 
x 32 SQUARED, + 4 x 32 +3.
SO THE MAXIMUM HEIGHT 
IS EQUAL TO 67 FEET.
AND THEN 
FOR THE THIRD QUESTION,
WE WANT TO DETERMINE 
HOW FAR HORIZONTALLY
DOES THE ARROW TRAVEL 
BEFORE HITTING THE GROUND.
WELL WHEN THE ARROW 
HITS THE GROUND,
THE VERTICAL HEIGHT 
WOULD BE 0,
WHICH MEANS Y WOULD BE 
EQUAL TO 0.
SO WE'RE GOING TO SET Y 
EQUAL TO 0
SO WE'D HAVE THE EQUATION
0 = -ONE SIXTEENTH X SQUARED 
+ 4X +3.
NOW THIS EQUATION 
IS NOT GOING TO FACTOR
SO WE'LL HAVE TO SOLVE THIS 
USING THE QUADRATIC FORMULA.
AND IT'S GOING TO BE EASIER 
IF WE ELIMINATE THIS FRACTION
BY MULTIPLYING BOTH SIDES 
BY 16
OR IF WE WANT "A" 
TO BE POSITIVE,
LET'S MULTIPLY BOTH SIDES 
BY A -16.
REMEMBER MULTIPLYING 
BOTH SIDES OF AN EQUATION
BY A CONSTANT IS NOT GOING 
TO CHANGE THE SOLUTIONS.
SO WE'LL HAVE THE 0=
THIS WOULD BE 
POSITIVE 1X SQUARED - 64X
AND THIS WOULD BE MINUS 48.
SO NOW WE'LL USE 
THE QUADRATIC FORMULA
WHERE "A" IS EQUAL TO 1, 
B IS EQUAL TO -64
AND C IS EQUAL TO -48.
SO WE'LL HAVE X = -B OR -64, 
THAT WILL BE POSITIVE 64
+ OR - THE SQUARE ROOT 
OF B SQUARED,
WHICH IS - 64 SQUARED -4 x 
"A"
WHICH IS 1 x C WHICH IS - 48.
HOWEVER 2 x A OR 2 x 1 
WHICH IS 2.
NOW LET'S GO AHEAD 
AND BEGIN TO SIMPLIFY THIS.
WE'LL HAVE X = 64 + OR - WE'LL 
COME BACK TO THE DISCRIMINATE
ALL THIS IS DIVIDED BY 2.
LET'S EVALUATE THIS 
ON THE CALCULATOR.
SO THE DISCRIMINATE IS - 64 
SQUARED - 4 x 1
WHICH WE DON'T NEED
BUT I'LL PUT IT IN THERE 
x - 48.
SO WE HAVE 4,288 
UNDERNEATH THE SQUARE ROOT
AND BECAUSE WE KNOW 
X HAS TO BE POSITIVE,
IT'S A HORIZONTAL DISTANCE,
WE ONLY HAVE TO BE CONCERNED
ABOUT THE POSITIVE SOLUTION 
HERE
WHICH WOULD BE 64 + 
THE SQUARE ROOT OF 4,288
ALL DIVIDED BY 2.
SO LET'S GO BACK 
TO THE CALCULATOR.
OUR NUMERATOR IS GOING TO BE 
64 + THE SQUARE ROOT OF 4,288.
NOTICE HOW WE HAVE 
TWO CLOSED PARENTHESIS HERE,
ONE FOR THE SQUARE ROOT 
AND ONE FOR THE NUMERATOR,
DIVIDED BY TWO.
SO THE ARROW TRAVELS
APPROXIMATELY 64.7 FEET 
HORIZONTALLY
BEFORE HITTING THE GROUND.
OKAY, HOPE THIS EXAMPLE HELPS.
