Question

A soccer ball is kicked toward the goal. The height of the ball is modeled by the function h(t) = −16t2 + 48t where t equals the time in seconds and h(t) represents the height of the ball at time t seconds. What is the axis of symmetry, and what does it represent?
t = 1.5; it takes 1.5 seconds to reach the maximum height and 3 seconds to fall back to the ground

t = 1.5; it takes 1.5 seconds to reach the maximum height and 1.5 seconds to fall back to the ground

t = 3; it takes 3 seconds to reach the maximum height and 3 seconds to fall back to the ground

t = 3; it takes 3 seconds to reach the maximum height and 6 seconds to fall back to the ground

I would appreciate it if whoever answers this also explains how the answer was reached, particularly in regards to latter half of the answers.

Answer 1

Thank you for asking what you really want to know.

The axis of symmetry is the line through the vertex that divides the graph into two parts that are mirror images of each other. Any axis of symmetry for any graph or solid object is a line such that everything you see on one side of the line mirrors what you see in the opposite direction on the other side of the line.

For the graph of a vertical parabola, there are several ways you can find the axis of symmetry:

• find the x-coordinate of the vertex (from a graph or by other means)

• use an equation that tells you the axis of symmetry

• locate two points at the same level on the graph and find their midpoint. (It is often convenient to use the two zeros for this.)

When the equation is of the form

... y = ax^2 + bx +c

the axis of symmetry is given by

... x = -b/(2a)

In your equation, you have a=-16 and b=48, so the axis of symmetry is

... t = -48/(2(-16)) = 48/32 = 3/2

... t = 1.5

Another way you can find the axis of symmetry for your graph is to factor the equation:

... h(t) = -16t(t -3)

You know from the zero product rule that h(t) = 0 when either t=0 or t-3=0. In the latter case, this means when t=3. These value of t (0 or 3) give the same value of h(t), namely zero. Then the third bullet point above applies and we can find the axis of symmetry as the midpoint between t=0 and t=3. That is t=(0+3)/2 = 3/2.

The symmetry aspect of this means if it took 1.5 seconds to go up, it will take 1.5 seconds to come down (the difference between 1.5 and 3).

Have a look at the graph to see a picture of this.

The appropriate choice among those offered is ...

... t = 1.5; it takes 1.5 seconds to reach the maximum height and 1.5 seconds to fall back to the ground