Lesson 3.3: Rates of Change and Behavior of Graphs

New Concept

This lesson explores how functions change. You'll calculate the average rate of change, $\frac{\Delta y}{\Delta x}$, to understand if a function is increasing or decreasing and use its graph to pinpoint key features like maximum and minimum values (extrema).

What’s next

Get ready to master these concepts! Next, you'll tackle worked examples, practice cards, and interactive graphs to sharpen your skills.

Property

A rate of change describes how an output quantity changes relative to the change in the input quantity. The units on a rate of change are “output units per input units.”
The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.

To calculate the average rate of change between two points, find the difference in the output values ($Δy$) and divide it by the difference in the input values ($Δx$).

Examples

• Find the average rate of change of $f(x) = x^2 + \frac{1}{x}$ on the interval $[1, 3]$. We compute $f(1)=2$ and $f(3)=\frac{28}{3}$. The average rate of change is $\frac{\frac{28}{3} - 2}{3 - 1} = \frac{\frac{22}{3}}{2} = \frac{11}{3}$.

• A plant's height is recorded as 5 cm in week 1 and 25 cm in week 5. The average growth rate is the change in height divided by the change in time: $\frac{25 - 5}{5 - 1} = \frac{20}{4} = 5$ cm per week.

Property

A function $f$ is an increasing function on an open interval if $f(b) > f(a)$ for any two input values $a$ and $b$ in the given interval where $b > a$.
A function $f$ is a decreasing function on an open interval if $f(b) < f(a)$ for any two input values $a$ and $b$ in the given interval where $b > a$. The average rate of change for an increasing function is positive, while it is negative for a decreasing function.

Examples

• The function $f(x) = x^2$ is decreasing on the interval $(-\infty, 0)$ because the graph goes down as you move from left to zero. It is increasing on $(0, \infty)$ because the graph goes up after zero.

• The function $f(x) = x^3$ is always increasing on its entire domain, $(-\infty, \infty)$, because the graph consistently rises from left to right.

Property

A function $f$ has a local maximum at $x = b$ if there exists an interval $(a, c)$ with $a < b < c$ such that, for any $x$ in the interval $(a, c)$, $f(x) \leq f(b)$.
Likewise, $f$ has a local minimum at $x = b$ if there exists an interval $(a, c)$ with $a < b < c$ such that, for any $x$ in the interval $(a, c)$, $f(x) \geq f(b)$.
These points are called local extrema and occur where a function changes from increasing to decreasing or vice versa.

Examples

• The function $f(x) = x^2$ has a local minimum value of $0$ at $x=0$. This is the bottom point of the U-shaped graph.

• The function $g(x) = -(x-3)^2 + 5$ has a local maximum value of $5$ at $x=3$. This is the peak of the inverted U-shape.

Property

The absolute maximum of $f$ at $x = c$ is $f(c)$ where $f(c) \geq f(x)$ for all $x$ in the domain of $f$.
The absolute minimum of $f$ at $x = d$ is $f(d)$ where $f(d) \leq f(x)$ for all $x$ in the domain of $f$.
Not every function has an absolute maximum or minimum value.

Examples

• The function $f(x) = x^2 + 1$ has an absolute minimum of $1$ at $x=0$. It has no absolute maximum because its graph extends upward to infinity.

• On the restricted domain $[0, 10]$, the function $g(x) = x$ has an absolute minimum of $0$ (at $x=0$) and an absolute maximum of $10$ (at $x=10$).