The height of a softball, in feet, $t$ seconds after someone throws it straight up, can be defined by $f(t) = \text-16t^2+32t+5$. The input of function $f$ is time, and the output is height.

We can find the output of this function at any given input. For instance:

• At the beginning of the softball's journey, when $t = 0$, its height is given by $f(0)$.
• Two seconds later, when $t=2$, its height is given by $f(2)$.

The values of $f(0)$ and $f(2)$ can be found using a graph or by evaluating the expression $\text-16t^2+32t+5$ at those values of $t$. What if we know the output of the function and want to find the inputs? For example:

• When does the softball hit the ground?

  Answering this question means finding the values of $t$ that make $f(t)=0$, or solving $\text -16t^2+32t+5=0$. 

• How long will it take the ball to reach 8 feet?

  This means finding one or more values of $t$ that make $f(t) = 8$, or solving the equation $\text -16t^2+32t+5=8$.

The equations $\text -16t^2+32t+5=0$ and $\text -16t^2+32t+5=8$ are quadratic equations. One way to solve these equations is by graphing $y = f(t)$.

• To answer the first question, we can look for the horizontal intercepts of the graph, where the vertical coordinate is 0.
• To answer the second question, we can look for the horizontal coordinates that correspond to a vertical coordinate of 8.

Often, when we are modeling a situation mathematically, an approximate solution is good enough. Sometimes, however, we would like to know exact solutions, and it may not be possible to find them using a graph. In this unit, we will learn more about quadratic equations and how to solve for exact answers using algebraic techniques.