Suppose the ball was at the origin. Use the vertex and the axis of symmetry to find how far the first kick traveled. For the second kick, write the function as y=a(x-p)(x-q). Use it to find the x-intercepts and the vertex. How can you interpret the results?
Farther: The second kick traveled farther before hitting the ground. Higher: The first kick traveled higher.
Practice makes perfect
Let's begin by studying the path of the first kick. We can suppose the ball was at the origin (0,0). Then, it was kicked and reached a maximum height of 8 yards when it was 20 yards away from the kicker.
The path of the ball has the shape of a parabola that opens downward, passes through the origin, and has its vertex at ( 20, 8). This means that its axis of symmetry is x= 8. Then, due to symmetry, it must intercept the x-axis again at (40,0). Using this, let's draw the entire path.
Therefore, the first kick traveled 40 yards away from the player. For the second kick, we know the path is represented by the function y=x(0.4-0.008x). Let's rewrite it to find the x-intercepts.
We can see our zeros are at 0 and 50, so we can claim that the second kick travels 50 yards before hitting the ground. Additionally, we can find the maximum height reached by the ball by finding the y-coordinate of the vertex.
Function
x-coordinate
y-coordinate
f(x)=a(x-p)(x-q)
x=p+q/2
y=f(p+q/2)
In our case, p=0 and q=50. Thus, x= 0+502=25 is the x-coordinate of the vertex. Let's find the y-coordinate by evaluating x=25 into the function.
We now know the second kick reached a maximum height of 5 yards when it was 25 yards away from the player.
Finally, by comparing all the data, we can conclude that the second kick traveled farther before hitting the ground, but the first kick traveled higher.
