7.14.2 Interpreting a Functional Relationship between Two Quantities

The Meaning of Values on a Graph

Let’s say a tennis ball is hit straight up in the air, and its height in feet above the ground is modeled by the equation $f(t)=4+12t-16t^2$ where $t$ represents the time in seconds after the ball is hit. Here is a graph that represents the function, from the time the tennis ball was hit until the time it reached the ground.

In the graph, we can see some information we already know, and some new information:

• The 4 in the equation means the graph of the function intersects the vertical axis at 4. It shows that the tennis ball was 4 feet off the ground at $t=0$, when it was hit.
• The horizontal intercept is $(1,0)$. It tells us that the tennis ball hit the ground 1 second after it was hit.
• The vertex of the graph is at approximately $(0.4,6.3)$. This means that about 0.4 second after the ball was hit, it reached the maximum height of about 6.3 feet.

The equation can be written in factored form as $f(t)=(-16t-4)(t-1)$. From this form, we can see that the zeros of the function are $t=1$ and $t=\frac{1}{4}$. The negative zero, -14, is not meaningful in this situation because the time before the ball was hit is irrelevant.

Try it

Try It: The Meaning of Values on a Graph

An object is thrown upward from a height of 5 feet with a velocity of 60 feet per second. Its height h(t) in feet after $t$ seconds is modeled by the function $h(t)=5+60t-16t^2$ and graphed below.

What is the value of the vertical intercept ($y$-intercept) and what does it mean in the function?

Here is how to determine the $y$-intercept:

The vertical intercept, or $y$-intercept, is where the graph crosses the $y$-axis. It occurs when $x=0$. You can also identify the $y$-intercept by looking at the constant in the function. This occurs at $y=5$. The point is (0,5). The $y$-intercept represents the initial height. It means the object is thrown from 5 feet above the ground.