Analyzing Graphs: Increasing/Decreasing Intervals and Concavity

Graphs of Functions

Understanding Increasing/Decreasing Intervals and Concavity

Analyzing the graph of a function is a key skill in Precalculus. It helps us understand where a function is rising or falling (increasing or decreasing), and where it is concave up or concave down. Points of inflection are where the concavity changes.

• Increasing Interval: The function rises as you move from left to right.

• Decreasing Interval: The function falls as you move from left to right.

• Concave Up: The graph is shaped like a cup (∪); the slope is increasing.

• Concave Down: The graph is shaped like a cap (∩); the slope is decreasing.

• Point of Inflection: The point where the graph changes concavity.

Given: The graph of g(x) has points of inflection at $x = -1$ and $x = 1$.

Identifying Intervals

• To find where the graph is increasing and concave up, look for intervals where the graph rises and is shaped like a cup (∪).

• To find where the graph is decreasing and concave up, look for intervals where the graph falls and is shaped like a cup (∪).

Example: Analyzing the Provided Graph

• Points of Inflection: $x = -1$ and $x = 1$ (where concavity changes).

• Concave Up: The graph is concave up on the interval $(-1, 1)$.

• Increasing: The graph increases on $(-3, 0)$ and $(2, 5)$ (approximate from the graph).

• Decreasing: The graph decreases on $(-5, -3)$, $(0, 2)$ (approximate from the graph).

Interval	Increasing/Decreasing	Concavity
$(-1, 1)$	Increasing on $(-1, 0)$ Decreasing on $(0, 1)$	Concave up

Answers to the Questions

• On what intervals is the graph of g increasing and concave up? Answer: $(-1, 0)$

• On what intervals is the graph of g decreasing and concave up? Answer: $(0, 1)$

Summary Table of Intervals

Interval	Behavior
$(-1, 0)$	Increasing & Concave Up
$(0, 1)$	Decreasing & Concave Up

Additional info: In calculus, these intervals are found by analyzing the first and second derivatives, but in Precalculus, we use the graph's shape and direction to determine them.