In the example below, we want to look at the inputs and outputs for a function and interpret what they tells us. In both examples, the function is a quadratic function that models the rise and fall of an object thrown in the air.

**Example 1** Suppose a ball is thrown into the air has its height (in feet) given by the function

$latex \displaystyle h(t)=6+128t-16{{t}^{2}}$

a. Find *h*(1) and explain what it means.

b. Find the height of the ball 4 seconds after it is thrown.

c. Test other values of to decide if the ball eventually falls. When does the ball stop climbing?

**Example 2** Suppose a ball is thrown into the air has its height (in feet) given by the function

$latex \displaystyle h(t)=6+96t-16{{t}^{2}}$

a. Find *h*(1) and explain what it means.

b. Find the height of the ball 3 seconds after it is thrown.

c. Test other values of to decide if the ball eventually falls. When does the ball stop climbing?

The key thing is to test the function at enough points to convince yourself that the peak is really the peak. If the peak height occurs at *x* = 4.2, will finding *h*(3), *h*(4), and *h*(5) be enough to find that peak? If we have a graph, we can use it to find the peak. But if we only have the function we need to fine tune the input to zoom in on wherever the peak is.

In each case, the values of *t* are in seconds and *h*(*t*) is in feet. We want to find an *h*(*t*) value that is higher then those on either side.