Quadratic Equation

Question

A golf ball is hit, and it lands at a point on the same horizontal plane 53m away. The path of the ball took it just over a 9m tall tree that was 8m in front of the golfer.

 

a) Assume the ball is hit from the origin of a coordinate plane. Find a quadratic function that describes the path of the ball.

 

b) What is the maximum height of the ball?

 

c) Is it possible to move the origin in this situation and develop another quadratic function to describe the path? If so, find a second quadratic function.

 

 

Help me!

Answer 1

An equation of the path is y = ax²+bx+c.

 

y is the height above the horizontal plane at a distance of x feet from the origin.  

 

The points (0,0) and (53,0) are on the graph.

 

Since (0,0) is on the curve, 0 = a(0)²+b(0)+c.  So, c = 0

 

We have y = ax²+bx

 

Since (8,9) and (53,0) are on the graph, we get:

 

       64a + 8b = 9

       2809a + 53b = 0  (divide by 53 to get 53a+b=0)

 

So, 64a + 8b = 9

     53a +   b = 0

 

From the second equation, b = -53a

 

So, 64a +8(-53a) = 9      -360a = 9   a = -1/40

 

                                                    b = -53(-1/40) = 53/40

 

The equation is y = (-1/40)x² + (53/40)x

 

Max height when x = -b/(2a) = -(53/40)/[-(1/20)] = 26.5

 

Max height = (-1/40)(26.5)²+(53/40)(26.5) ≈ 17.6 feet

 

Answer 2

Use the graphing form for the quadratic equation:

 

y = a(x - h)² + k

 

(h,k) is the vertex,

 

a is the stretch factor

 

you must replace a, h, and k with numbers

 

since the parabola goes from 0 to 53 on the x-axis, the 

vertex will be at ( 26.5, k)

 

Use the points (8,9) and (53,0)

 

9 = a(8 - 26.5)² + k;  0 = a(53 - 26.5)²  + k

 

9 = 342.25a + k

 

0 = 702.25a + k

 

subtract the first equation from the second: 

 

-9 = 360a

 

-1/40 = a

 

k = -702.25a = 17.57

 

y = (-1/40) (x - 26.25)² + 17.57

 

b) maximum height is y value of vertex = 17.57 m

 

c) move the y-axis to the right until it is the axis of symmetry for the flight of the golf ball;

    you then would have the following points: vertex (0, k), (26.5, 0), (-26.5, 0) and (-18.5, 9)

 

Can you work it from there?  If not, let me know and I will help you.