We have two different viewpoints when examining functions.

$\blacktriangleright$ Numbers: Functions consist of individual pairs of individual numbers. When we evaluate functions, we think of each pair one at a time. The graphs of functions are individual dots plotted to represent points, which visually encode the function pairs.

$\blacktriangleright$ Intervals: As we compare function values, we notice patterns, which we call behavior. We naturally group many pairs together where the function has similar behavior. Collecting pairs forms domain intervals and we begin thinking in terms of intervals.

In particular, we calculate rate of change over intervals, which measures behavior.

The rate of change of the function $f$ over the interval $[a, b]$ is given by $\frac {f(b) - f(a)}{b - a}$.

However, intervals are just collections of individual numbers. Is there a way to translate this rate of change idea down to single numbers?

The answer is yes.

We will begin this thought here with quadratic functions. We’ll extend this thought a bit throughout the course. Then Calculus will answer the question fully with the derivative.

Learning Outcomes

In this section, students will

• develop the $iRoC$ function.
• analyze quadratic functions.