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

Equations & Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the power of one. Solving these equations involves isolating the variable on one side of the equation.

• Key Point 1: To solve a linear equation, use inverse operations to isolate the variable.

• Key Point 2: Always simplify both sides of the equation as much as possible before solving.

• Example: Solve $8n + 8 = 8n - 4$ Subtract $8n$ from both sides: $8 = -4$ (No solution; this is a contradiction.)

Solving for a Variable

Sometimes, equations involve more than one variable, and you may be asked to solve for a specific variable in terms of others.

• Key Point 1: Rearrange the equation to isolate the desired variable.

• Example: Solve $2x - 3y = 8$ for $y$: $2x - 3y = 8$ $-3y = 8 - 2x$ $y = \frac{2x - 8}{3}$

Solving Formulas

Formulas are equations that express relationships between variables. You may need to solve a formula for a particular variable.

• Key Point 1: Use algebraic manipulation to isolate the variable of interest.

• Example: Given $C = \frac{5}{9}(F - 32)$, solve for $F$: $C = \frac{5}{9}(F - 32)$ $9C = 5(F - 32)$ $9C = 5F - 160$ $5F = 9C + 160$ $F = \frac{9C + 160}{5}$

Graphs of Equations

Graphing Linear Equations

Linear equations can be graphed as straight lines on the coordinate plane. The general form is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Key Point 1: The slope ($m$) indicates the steepness and direction of the line.

• Key Point 2: The y-intercept ($b$) is the point where the line crosses the y-axis.

• Example: For $y = 3x + 6$, the slope is $3$and the y-intercept is $6$.

Finding Slope and Intercept

The slope of a line through two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

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

• Example: Find the slope of the line passing through $(2, 3)$ and $(4, 7)$: $m = \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2$

Types of Slope

• Positive Slope: Line rises from left to right.

• Negative Slope: Line falls from left to right.

• Zero Slope: Horizontal line.

• Undefined Slope: Vertical line.

Writing Equations of Lines

Given a point $(x_1, y_1)$ and a slope $m$, the equation of the line is:

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Slope-Intercept Form: $y = mx + b$

• Example: Find the equation of the line with slope $2$passing through$(1, 3)$:$ y - 3 = 2(x - 1) y = 2x + 1 $

Functions

Definition of a Function

A function is a relation in which each input (domain value) corresponds to exactly one output (range value).

• Key Point 1: Functions can be represented by equations, tables, or graphs.

• Key Point 2: The vertical line test determines if a graph represents a function: if any vertical line crosses the graph more than once, it is not a function.

• Example: The equation $y = x^2$ is a function, but $x^2 + y^2 = 1$ (a circle) is not a function of $x$.

Domain and Range

The domain of a function is the set of all possible input values, and the range is the set of all possible output values.

• Key Point 1: For rational functions, exclude values that make the denominator zero.

• Key Point 2: For square root functions, the expression under the root must be non-negative.

• Example: For $f(x) = \frac{x+5}{x-5}$, the domain is all real numbers except $x = 5$.

Evaluating Functions

To evaluate a function, substitute the given value for the variable and simplify.

• Example: If $f(x) = x^2 - 4$, then $f(2) = 2^2 - 4 = 0$.

Applications of Linear Equations

Word Problems and Modeling

Linear equations are used to model real-world situations, such as pricing, salaries, and cost analysis.

• Key Point 1: Translate the problem into an equation using variables to represent unknowns.

• Key Point 2: Solve the equation to answer the question posed by the problem.

• Example: If a store marks up wholesale price by 15%, the retail price $R$ is $R = 1.15W$, where $W$ is the wholesale price.

Tables

Table: Types of Slope

Slope Value	Type	Graph Description
Positive	Rising	Line goes up left to right
Negative	Falling	Line goes down left to right
Zero	Horizontal	Flat line
Undefined	Vertical	Vertical line

Table: Function vs. Not a Function (from Table Data)

Domain	Range	Function?
1	11	Yes
-1	17	Yes
1	13	No (repeats domain value)

Additional info: In a function, each domain value must correspond to only one range value.

Summary

• Solving equations and inequalities is foundational for algebraic reasoning.

• Graphing and interpreting linear equations helps visualize relationships between variables.

• Understanding functions, their domains, and ranges is crucial for further study in mathematics.

• Applications connect algebraic concepts to real-world scenarios.