College Algebra: Functions, Graphs, and Equations Study Guide

Functions and Their Graphs

Definition of a Function

A function is a relation that assigns exactly one output value for each input value. In algebra, functions are often represented as equations, tables, or graphs.

• Domain: The set of all possible input values (usually x-values).

• Range: The set of all possible output values (usually y-values).

• Function Notation: $f(x)$ represents the value of the function at $x$.

Example: The function $f(x) = 2x + 3$ assigns to each $x$ the value $2x + 3$.

Graphing Functions

Graphs visually represent the relationship between input and output values. Common types include linear, quadratic, and square root functions.

• Linear Function: $f(x) = mx + b$ (straight line)

• Quadratic Function: $f(x) = ax^2 + bx + c$ (parabola)

• Square Root Function: $f(x) = \sqrt{x}$ (curve starting at $x=0$)

Example: The graph of $f(x) = x^2$ is a parabola opening upwards.

Types of Equations and Their Solutions

Linear Equations

Linear equations are equations of the first degree, meaning the highest power of the variable is one.

• General Form: $ax + b = 0$

• Solution: $x = -\frac{b}{a}$

Example: Solve $2x - 4 = 0$:

• $2x = 4$

• $x = 2$

Quadratic Equations

Quadratic equations have the form $ax^2 + bx + c = 0$.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Factoring: Express as a product of binomials and set each factor to zero.

Example: Solve $x^2 - 5x + 6 = 0$:

• Factoring: $(x-2)(x-3) = 0$

• Solutions: $x = 2$, $x = 3$

Square Root Equations

Equations involving square roots require isolating the radical and squaring both sides to solve.

• General Form: $\sqrt{x} = a$

• Solution: $x = a^2$

Example: Solve $\sqrt{x} = 5$:

• $x = 25$

Interpreting Graphs

Identifying Functions from Graphs

To determine if a graph represents a function, use the vertical line test: if any vertical line crosses the graph more than once, it is not a function.

• Linear Graphs: Always pass the vertical line test.

• Parabolas: $y = x^2$ passes the test; $x = y^2$ does not.

• Piecewise Functions: May have different rules for different intervals.

Example: The graph of $y = |x|$ is a function because it passes the vertical line test.

Matching Equations to Graphs

Recognizing the shape and key features of graphs helps match them to their equations.

• Intercepts: Where the graph crosses the axes.

• Vertex: The turning point of a parabola.

• End Behavior: How the graph behaves as $x \to \infty$ or $x \to -\infty$.

Example: The graph opening upwards with vertex at $(0,0)$ matches $y = x^2$.

Properties of Functions

Domain and Range

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

• Domain of $f(x) = \sqrt{x}$: $x \geq 0$

• Range of $f(x) = x^2$: $y \geq 0$

Example: For $f(x) = \sqrt{x-2}$, domain is $x \geq 2$.

Summary Table: Common Functions and Their Graphs

Function	Equation	Graph Shape	Domain	Range
Linear	$y = mx + b$	Straight line	All real numbers	All real numbers
Quadratic	$y = ax^2 + bx + c$	Parabola	All real numbers	$y \geq$ vertex (if $a > 0$)
Square Root	$y = \sqrt{x}$	Curve starting at $x=0$	$x \geq 0$	$y \geq 0$
Absolute Value	$y = |x|$	V-shaped	All real numbers	$y \geq 0$

Additional info:

• Some questions in the file involve matching equations to graphs, identifying domains/ranges, and solving equations involving square roots and quadratics.

• Graph images suggest focus on function identification, graph interpretation, and equation solving, all central to College Algebra.