3.3 Rates of Change and Behavior of Graphs

Rates of Change

Average Rate of Change

The average rate of change tells you how much a function's output changes per unit of input over a specific interval. It's calculated with this formula:

$\frac{f(b) - f(a)}{b - a}$

• $a$ is the starting input value and $b$ is the ending input value
• $f(a)$ and $f(b)$ are the corresponding output values
• Geometrically, this is the slope of the secant line connecting the points $(a, f(a))$ and $(b, f(b))$ on the graph

Example: Suppose $f(x) = x^2$. The average rate of change from $x = 1$ to $x = 4$ is:

$\frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = 5$

This means that on average, the function increases by 5 units in $y$ for every 1 unit increase in $x$ over that interval.

Interpreting the sign:

• Positive value: the function increases on average over the interval (output goes up as input goes up)
• Negative value: the function decreases on average over the interval (output goes down as input goes up)
• Zero value: the function has the same output at both endpoints, so there's no net change over the interval

Keep in mind that a zero average rate of change doesn't necessarily mean the function was flat the whole time. It could have gone up and come back down.

Intervals of Increasing, Decreasing, and Constant Behavior

When describing where a function increases or decreases, you always read the graph from left to right and express the intervals using $x$-values (input values), not $y$-values.

• Increasing interval: As $x$ moves to the right, $f(x)$ rises. For any two points in the interval, if $x_1 < x_2$, then $f(x_1) < f(x_2)$.
• Decreasing interval: As $x$ moves to the right, $f(x)$ falls. If $x_1 < x_2$, then $f(x_1) > f(x_2)$.
• Constant interval: $f(x)$ stays the same no matter which $x$-values you pick in the interval. The graph is a horizontal segment.

A common mistake is writing the $y$-values as the interval. If a function increases from the point $(1, 3)$ to $(4, 10)$, the increasing interval is $(1, 4)$, not $(3, 10)$.

Behavior of Graphs

Local Extrema

A local maximum is a point where the function value is higher than all nearby points. A local minimum is a point where the function value is lower than all nearby points. Together, these are called local extrema.

• A local maximum occurs where the function switches from increasing to decreasing. Picture the top of a hill.
• A local minimum occurs where the function switches from decreasing to increasing. Picture the bottom of a valley.

When you identify a local extremum, report it as a $y$-value (the function value), and state where it occurs as an $x$-value. For example: "There is a local maximum of 7 at $x = 3$."

A function can have multiple local maxima and minima, or none at all. A straight line, for instance, has no local extrema because it never changes direction.

Absolute Extrema

Absolute extrema are the single highest and single lowest values a function achieves over its entire domain.

• The absolute maximum is the largest $y$-value anywhere on the graph. No other point reaches higher.
• The absolute minimum is the smallest $y$-value anywhere on the graph. No other point dips lower.

Not every function has absolute extrema. A function like $f(x) = x^2$ has an absolute minimum at $(0, 0)$ but no absolute maximum because it keeps growing forever. Functions defined on a closed interval $[a, b]$ will always have both an absolute max and an absolute min.

How to find absolute extrema from a graph:

1. Identify all local maxima and local minima on the graph.
2. Check the function values at the endpoints of the domain (if the domain is a closed interval).
3. Compare all of these $y$-values. The greatest is the absolute maximum; the least is the absolute minimum.