Question

a game requires players to throw a ball into a target. the table below shows values from the function that represents one path of a thrown ball, where x represents the horizontal distance the ball travels in feet and f(x) represents the height of the ball in feet.

x	f(x)
0.25	13
0.5	20
0.75	25
1	28
1.25	29
1.5	28

if the ball enters the target at the highest point in its path, based on the table, what is the height of the target?

• 1.5 feet
• 4 feet
• 28 feet
• 29 feet

Explanation:

Step1: Identify the vertex pattern

The ball's path is a parabola (since it's a projectile motion, symmetric). We look for the maximum value of \( f(x) \) as the highest point.

Step2: Analyze the table values

Check the \( f(x) \) values: at \( x = 1.25 \), \( f(x)=29 \); at \( x = 1 \), \( f(x)=28 \); at \( x = 1.5 \), \( f(x)=28 \). Before \( x = 1.25 \), \( f(x) \) was increasing (from 4 to 29), after \( x = 1.25 \), it starts decreasing (29 to 28). So the highest point is at \( x = 1.25 \) with \( f(x)=29 \)? Wait, no, wait. Wait, wait, let's re - check. Wait, when \( x = 1.25 \), \( f(x)=29 \), then at \( x = 1.5 \), it's 28. Before \( x = 1.25 \): at \( x = 0 \), 4; \( x = 0.25 \),13; \( 0.5 \),20; \( 0.75 \),25; \( 1 \),28; \( 1.25 \),29. After \( x = 1.25 \), \( x = 1.5 \),28. So the function increases up to \( x = 1.25 \) and then decreases. So the maximum (highest point) is at \( x = 1.25 \), \( f(x)=29 \)? But wait, the options: 1.5, 4, 28, 29. Wait, but let's check the symmetry. The axis of symmetry of a parabola is the mid - point between the points where the function has the same value. Notice that at \( x = 1 \), \( f(x)=28 \) and at \( x = 1.5 \), \( f(x)=28 \). The mid - point of \( x = 1 \) and \( x = 1.5 \) is \( x=\frac{1 + 1.5}{2}=\frac{2.5}{2}=1.25 \). So the vertex (highest point) is at \( x = 1.25 \), and \( f(1.25)=29 \). But wait, the question says "the ball enters the target at the highest point in its path". So the height of the target is the value of \( f(x) \) at the highest point.

Answer:

29 feet