College Algebra: Functions, Graphs, and Applications Study Guide

Functions and Their Representations

Definition and Examples of Functions

A function is a relation in which each input (usually x) has exactly one output (usually y). Functions can be represented by equations, graphs, tables, or sets of ordered pairs.

• Example 1: $f(x) = \frac{x-2}{x+3}$

• Example 2: $g(x) = \sqrt{6-2x}$

• Example 3: $h(x) = 2x^2 - 2$

Functions may be linear, quadratic, radical, or other types, each with distinct properties and graphs.

Determining Whether a Relation is a Function

To determine if a relation is a function, check if each input corresponds to only one output. The vertical line test is commonly used for graphs: if any vertical line crosses the graph more than once, it is not a function.

• Graphical Relations: Analyze provided graphs using the vertical line test.

• Set of Ordered Pairs: $\{(2,5), (3,6), (2,7), (3,8)\}$ is not a function because the input 2 maps to both 5 and 7.

• Equations: $y = x^2 + 2x - 6$ and $y^2 = 5 - x$ (the latter is not a function since some x-values yield two y-values).

Graphical Analysis of Functions

Increasing, Decreasing, and Constant Intervals

The behavior of a function can be described by intervals where it is increasing, decreasing, or constant. This is determined by analyzing the graph:

• Increasing: The function rises as x increases.

• Decreasing: The function falls as x increases.

• Constant: The function remains unchanged as x increases.

Use the graph to identify these intervals by observing the slope of the curve.

Difference Quotient

Definition and Application

The difference quotient is a fundamental concept in calculus and algebra, used to measure the average rate of change of a function:

• Formula:

$\frac{f(x+h) - f(x)}{h}$

• Apply this formula to a given function, such as $f(x) = 2x^2 - 6$.

• Example: Compute $\frac{f(x+h) - f(x)}{h}$ for $f(x) = 2x^2 - 6$.

Transformations of Functions

Graphing Transformed Functions

Transformations include shifts, stretches, and reflections. For example, $g(x) = f(x-3) + 3$ shifts the graph of $f(x)$ right by 3 units and up by 3 units.

• Horizontal Shift: $f(x-a)$ shifts right by $a$ units.

• Vertical Shift: $f(x) + b$ shifts up by $b$ units.

Piecewise Functions

Evaluating Piecewise Functions

A piecewise function is defined by different expressions over different intervals of the domain.

• Example:

$w(x) = \begin{cases} 3x + 5 & x < -1 \\ x^2 - 4 & -1 \leq x \leq 3 \\ 1 \times (x - 3) & 3 < x \end{cases}$

• To evaluate $w(x)$ at specific values, determine which interval the input falls into and use the corresponding expression.

Linear Models and Applications

Writing Salary as a Function

Linear functions can model real-world scenarios, such as salary based on commission:

• Example: If Caroline earns a base pay of $130 per week plus $9 for every item sold, her salary$S$as a function of the number of items$n$ sold is:

$S(n) = 130 + 9n$

Equations of Lines

Finding the Equation of a Line Through Two Points

The equation of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ can be found using the point-slope form:

• Formula:

$y - y_1 = m(x - x_1)$

• Where $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Example: Find the equation for points $(5, -2)$ and $(2, -2)$.

Quadratic Functions and Their Properties

Vertex, Intercepts, and Axis of Symmetry

Quadratic functions have the general form $f(x) = ax^2 + bx + c$. Key properties include:

• Vertex: The highest or lowest point, found at $x = -\frac{b}{2a}$.

• Y-intercept: The point where $x = 0$.

• X-intercepts: Solutions to $f(x) = 0$.

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$.

Example: For $f(x) = (x+4)^2 - 3$:

• Vertex: $(-4, -3)$

• Axis of symmetry: $x = -4$

Polynomial Zeros

Finding Real and Non-Real Zeros

To find the zeros of a polynomial, set $f(x) = 0$ and solve for $x$. For higher-degree polynomials, use factoring, the Rational Root Theorem, or synthetic division.

• Example: $f(x) = x^4 + 3x^2 + 4x + 12$

• Find all real and non-real zeros by factoring or using the quadratic formula as needed.

Tables

Piecewise Function Table (Inferred)

The following table summarizes the intervals and corresponding expressions for the piecewise function $w(x)$:

Interval	Expression
$x < -1$	$3x + 5$
$-1 \leq x \leq 3$	$x^2 - 4$
$3 < x$	$x - 3$

Additional info:

• Some graphs and tables were inferred based on standard College Algebra topics.

• All equations are provided in LaTeX format for clarity.