Question

Nick is practicing kicking field goals. His first kick is modeled by the parabola that passes through (0,0), (1,6.86), and (2,3) where x is the number of seconds that the ball is in the air, and y is the height of the ball in yards.

Answer 1

6.3 yards high

Explanation:

The three given points are (0,0), (1,6.86), and (2,3). Use these values in f(x) = ax2 + bx + c.

For (0,0), 0 = 0 + 0 + c.

So, c = 0.

For (1,6.86), 6.86 = a + b + c. Since c = 0:

a + b = 6.86(equation 1)

For (2,3), 3 = 4a + 2b + c. Since c = 0:

4a + 2b = 3(equation 2)

Multiply equation 1 by 2, and subtract the result from equation 2:

2a + 2b = 13.72

4a + 2b - (2a + 2b) = 3 - 13.72

2a = -10.72

a = - 5.36

Plug this value into equation 1:

-5.36 + b = 6.86

b = 12.22

So the function is f(x) = -5.36x2 + 12.22x.

When the ball hits the ground, the height is 0, or f(x) = 0:

Use the function to determine the height of the ball, f(x), after x = 1.5 seconds.

f(x) = -5.36x2 + 12.22x

= -5.36(1.5)2 + 12.22(1.5)

= 6.27

0 = -5.36x2 + 12.22x

0 ≈ x − 2.27(Divide both sides by -5.36x.)

x ≈ 2.27 ≈ 2.3

So, the ball will be about 6.3 yards high after 1.5 seconds.