College Algebra: Functions, Graphs, and Applications – Study Guide

Functions and Their Graphs

Definition and Representation of Functions

A function is a relation in which each input (from the domain) is assigned to exactly one output (in the range). Functions can be represented in various ways, including equations, tables, graphs, and verbal descriptions.

• Domain: The set of all possible input values (x-values) for which the function is defined.

• Range: The set of all possible output values (y-values) that the function can produce.

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

• Example: The equation $y = x^2$ defines a function because for every x, there is exactly one y.

Symmetry of Graphs

Understanding symmetry helps in graphing and analyzing functions.

• Symmetry with respect to the y-axis: If replacing x with -x yields the same equation, the graph is symmetric about the y-axis (even function).

• Symmetry with respect to the x-axis: If replacing y with -y yields the same equation, the graph is symmetric about the x-axis (not a function).

• Symmetry with respect to the origin: If replacing x with -x and y with -y yields the same equation, the graph is symmetric about the origin (odd function).

• Example: $y = x^3$ is symmetric about the origin (odd function).

Intercepts and Graphing Techniques

Intercepts are points where the graph crosses the axes.

• x-intercept: Set $y = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $y$.

• Graphing Steps:

  1. Find intercepts.

  2. Test for symmetry.

  3. Plot additional points as needed.

  4. Draw the curve smoothly through the points.

• Example: For $y = x^2 - 4$, the y-intercept is $-4$ and the x-intercepts are $x = 2$ and $x = -2$.

Equations of Lines

Standard and General Forms

Lines can be represented in several forms, each useful in different contexts.

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Point-Slope Form: $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line.

• General Form: $Ax + By + C = 0$, where $A$, $B$, and $C$ are constants.

• Parallel Lines: Have the same slope, $m_1 = m_2$.

• Perpendicular Lines: Slopes are negative reciprocals, $m_1 = -1/m_2$.

• Example: Find the equation of a line passing through $(2, 3)$ with slope $-4$: $y - 3 = -4(x - 2)$.

Relations and Functions

Difference Between Relations and Functions

A relation is any set of ordered pairs. A function is a special type of relation where each input has exactly one output.

• Vertical Line Test: Used to determine if a graph represents a function.

• Example: The relation $y^2 = x$ is not a function because for some $x$, there are two possible $y$ values.

Function Notation and Evaluation

Function notation is written as $f(x)$, which denotes the output of function $f$ for input $x$.

• Evaluating Functions: Substitute the input value into the function.

• Example: If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$.

Domain and Range

Finding the domain and range is essential for understanding the behavior of functions.

• Domain: All real numbers $x$ for which the function is defined (no division by zero, no square roots of negative numbers, etc.).

• Range: All possible output values $y$.

• Example: For $f(x) = \frac{1}{x-2}$, the domain is $x \neq 2$.

Average Rate of Change

Definition and Calculation

The average rate of change of a function between $x = a$ and $x = b$ is the change in the function's value divided by the change in $x$.

• Formula: $\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}$

• Example: For $f(x) = x^2$, the average rate of change from $x = 1$ to $x = 3$ is $\frac{f(3) - f(1)}{3 - 1} = \frac{9 - 1}{2} = 4$.

Increasing, Decreasing, and Constant Intervals

Definitions

A function can be classified as increasing, decreasing, or constant on certain intervals.

• Increasing: $f(x_2) > f(x_1)$ for $x_2 > x_1$ in the interval.

• Decreasing: $f(x_2) < f(x_1)$ for $x_2 > x_1$ in the interval.

• Constant: $f(x_2) = f(x_1)$ for all $x_2, x_1$ in the interval.

• Example: $f(x) = x^2$ is decreasing on $(-\infty, 0]$ and increasing on $[0, \infty)$.

Maximum and Minimum Values

Definitions

Functions may have local (relative) or absolute (global) maximum and minimum values.

• Local Maximum: The highest point in a small neighborhood.

• Local Minimum: The lowest point in a small neighborhood.

• Absolute Maximum: The highest point over the entire domain.

• Absolute Minimum: The lowest point over the entire domain.

• Example: $f(x) = -x^2$ has an absolute maximum at $x = 0$.

Practice Problems and Applications

Sample Problems

• Find the equation of a line given two points or a point and a slope.

• Determine the domain and range of various functions.

• Test whether a relation is a function using the vertical line test.

• Calculate the average rate of change for a given function over a specified interval.

• Identify intervals where a function is increasing, decreasing, or constant.

• Find local and absolute extrema (maximum and minimum values) of a function.

Table: Forms of Linear Equations

Form	Equation	Key Features
Slope-Intercept	$y = mx + b$	Slope $m$, y-intercept $b$
Point-Slope	$y - y_1 = m(x - x_1)$	Point $(x_1, y_1)$, slope $m$
General	$Ax + By + C = 0$	Standard form, useful for finding intercepts

Additional info:

• Some content inferred from standard College Algebra curriculum to provide context and completeness.

• Practice problems and solutions are referenced but not fully reproduced here; students should attempt these for mastery.