- 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.