More on Functions and Their Graphs: Increasing/Decreasing Intervals, Extrema, and Symmetry

Increasing, Decreasing, and Constant Functions

Definitions and Graphical Interpretation

Understanding how a function behaves on an interval is fundamental in precalculus. A function can be classified as increasing, decreasing, or constant on an open interval based on the relationship between its input and output values.

• Increasing on an interval I: For any $x_1 < x_2$ in $I$, $f(x_1) < f(x_2)$.

• Decreasing on an interval I: For any $x_1 < x_2$ in $I$, $f(x_1) > f(x_2)$.

• Constant on an interval I: For any $x_1, x_2$ in $I$, $f(x_1) = f(x_2)$.

When describing intervals of increase, decrease, or constancy, always use open intervals (parentheses), and read the graph from left to right. The intervals are given in terms of x-values, indicating where the function rises, falls, or remains unchanged.

Example: For the graph below, the function is:

• Decreasing on $(-2, 0)$

• Constant on $(0, 2)$

• Increasing on $(2, 4)$

Relative Maxima and Minima

Definitions and Identification

A function may attain local high or low points, called relative maxima and relative minima, within an open interval. These points are important for analyzing the overall behavior of the function.

• Relative Maximum at $x = a$: $f(a)$ is a relative maximum if there exists an open interval containing $a$ such that $f(a) > f(x)$ for all $x \neq a$ in the interval.

• Relative Minimum at $x = b$: $f(b)$ is a relative minimum if there exists an open interval containing $b$ such that $f(b) < f(x)$ for all $x \neq b$ in the interval.

• The maximum or minimum value is the y-value; the location is the corresponding x-value.

Example: Identify the numbers at which $f$ has a relative maximum and minimum, and state these values.

Tests for Symmetry

Definition and Algebraic Tests

Symmetry in graphs helps classify functions and predict their behavior. There are three main types of symmetry:

• y-axis symmetry: For every $(x, y)$ on the graph, $(-x, y)$ is also on the graph. Test by replacing $x$ with $-x$ in the equation; if the equation is unchanged, the graph is symmetric about the y-axis.

• x-axis symmetry: For every $(x, y)$ on the graph, $(x, -y)$ is also on the graph. Test by replacing $y$ with $-y$ in the equation; if unchanged, the graph is symmetric about the x-axis.

• Origin symmetry: For every $(x, y)$ on the graph, $(-x, -y)$ is also on the graph. Test by replacing $x$ with $-x$ and $y$ with $-y$; if unchanged, the graph is symmetric about the origin.

Example: Determine algebraically whether $y = x^2 - 1$ is symmetric with respect to the y-axis, x-axis, or origin.

Even and Odd Functions

Definitions and Properties

Functions can be classified as even, odd, or neither based on their symmetry properties:

• Even function: $f(-x) = f(x)$ for all $x$ in the domain. The graph is symmetric with respect to the y-axis.

• Odd function: $f(-x) = -f(x)$ for all $x$ in the domain. The graph is symmetric with respect to the origin.

• To determine if a function is even, odd, or neither, evaluate $f(-x)$ and compare to $f(x)$.

Example: Determine whether the following functions are even, odd, or neither:

• $h(x) = x^5 + 1$

• $q(x) = x^3 + 3x$

Applications and Practice

Analyzing Graphs for Function Properties

Use the graph of a function to answer questions about intervals of increase/decrease, relative extrema, and symmetry. For example, given a polynomial graph, you may be asked to:

• Determine intervals where $f$ is increasing or decreasing

• Find values of $x$ for which $f(x) < 0$, $f(x) = 0$, or $f(x) \geq 0$

• Identify relative maxima and minima

• Classify the function as even, odd, or neither

Example: For the function $f(x) = (x + 3)(x - 1)^2(x - 2)$, analyze the graph to answer the following:

• Intervals of increase and decrease

• Relative maxima and minima

• Values of $x$ for which $f(x) < 0$, $f(x) = 0$, $f(x) \geq 0$

• Symmetry properties

Domain, Range, and Intercepts

Given a graph, you may also be asked to determine:

• Domain: The set of all possible $x$-values for which $f(x)$ is defined.

• Range: The set of all possible $y$-values that $f(x)$ attains.

• x-intercepts: Points where the graph crosses the $x$-axis ($f(x) = 0$).

• y-intercept: The point where the graph crosses the $y$-axis ($x = 0$).