College Algebra: Fundamental Concepts, Equations, Functions, and Graphs

Section P.1: Algebraic Expressions

Evaluating Algebraic Expressions

Algebraic expressions use letters to represent numbers, called variables. Expressions may include addition, subtraction, multiplication, division, exponents, and roots.

• Exponential notation: If n is a counting number, $a^n = a \cdot a \cdot \ldots \cdot a$ (n times).

• Evaluating expressions: Substitute the given value for the variable and follow the order of operations.

Order of Operations

1. Parentheses

2. Exponents

3. Multiplication/Division (left to right)

4. Addition/Subtraction (left to right)

Example: Evaluate $3x - 2y$ for $x = 3$ and $y = 4$.

Example: Evaluate $x^2 + 3x - 12$ for $x = 8$.

Example: Evaluate $6 + 5(x - 6)^2$ for $x = 6$.

Example: Evaluate $\frac{4(x-6)}{2x-4}$ for $x = 8$.

Section P.1: Ordering Real Numbers

Using Inequality Symbols

Real numbers are graphed on a number line. Numbers increase from left to right. Use inequality symbols (<, >) to express order.

• Example: $-3 < 6$

• Example: $4 > 7$

• Example: $6 < 2$

Absolute Value

The absolute value of a real number $x$, denoted $|x|$, is its distance from 0 on the number line. Always nonnegative.

• Algebraic definition: $|x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$

• Example: $|-8| = 8$

• Example: $\left|\frac{-3}{11}\right| = 0.5$

Section P.1: Absolute Value Notation

Rewriting Absolute Value Expressions

Rewrite expressions without absolute value bars using the definition above.

• Example: $|-5| = 5$

• Example: $|8-x|$

• Example: $|y|-|x|$ for $x=6$, $y=8$

Section 1.1: Graphs

Rectangular Coordinate System

The coordinate system consists of horizontal (x-axis) and vertical (y-axis) lines. The intersection is the origin $(0,0)$. Points are plotted as $(x, y)$.

• The plane is divided into four quadrants.

• Example: Plot $(2,3)$, $(-5,7)$, $(0,-8)$, $(4,-6)$, $(4,0)$, $(-3,-5)$.

Intercepts

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

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

Section 1.1: Graphing Absolute Value

Basic Absolute Value Function

The function $f(x) = |x|$ is piecewise defined:

• $f(x) = x$ if $x \geq 0$

• $f(x) = -x$ if $x < 0$

The graph is a "V" shape.

x	y=2|x|	y=|x|-3
-3	6	0
-2	4	-1
-1	2	-2
0	0	-3
1	2	-2
2	4	-1
3	6	0

Section P.4: Operations with Polynomials

Addition and Subtraction

Polynomials are added/subtracted by combining like terms.

• Example: $(3x^2 - 4x^2 + 19x - 23) - (12x^2 - 3x^2 + 27x - 14)$

Multiplication

• Use distributive property and properties of exponents.

• Example: $(-9x^4)(4x^4)$

• Example: $2x^2(x^3 - x^2 + 2x - 8)$

FOIL Method for Binomials

Multiply $(a+b)(c+d)$ by distributing each term:

• First: $a \cdot c$

• Outside: $a \cdot d$

• Inside: $b \cdot c$

• Last: $b \cdot d$

Section 1.2: Solving Linear Equations

Linear Equations in One Variable

A linear equation can be written as $ax + b = 0$.

1. Simplify by distributing/combining like terms.

2. Collect variable terms on one side.

3. Add/subtract terms from both sides.

4. Multiply/divide both sides by the same number.

Example: $2(x-1) - 3x = x - 3(x-1)$

Section 1.7: Interval Notation and Linear Inequalities

Interval Notation

Represents solution sets on the real number line.

Interval Notation	Set-Builder Notation	Graph
(a, b)	{x | a < x < b}	Open endpoints
[a, b]	{x | a ≤ x ≤ b}	Closed endpoints
(a, b]	{x | a < x ≤ b}	Left open, right closed
[a, b)	{x | a ≤ x < b}	Left closed, right open
(a, ∞)	{x | x > a}	Open endpoint, arrow right
[a, ∞)	{x | x ≥ a}	Closed endpoint, arrow right
(-∞, b)	{x | x < b}	Arrow left, open endpoint
(-∞, b]	{x | x ≤ b}	Arrow left, closed endpoint

Solving Linear Inequalities

• Isolate the variable using properties of inequalities.

• Additive property: Add/subtract same number to both sides.

• Multiplication property: Multiply/divide both sides by positive number (direction unchanged); by negative number (reverse direction).

• Example: $3x - 5 > 4$

Section 2.1: Basics of Functions and Their Graphs

Domain and Range

The domain is the set of all possible input values; the range is the set of all possible output values.

• Example: Find domain and range of relation: {(0,5), (1,-2), (4,2), (8,9)}

Functions

A function assigns each input exactly one output. Use the vertical line test to determine if a graph is a function.

• Example: {(0,3), (1,2), (2,4), (5,6), (9,9)}

Function Notation

$f(x)$ denotes the value of function $f$ at $x$.

• Example: $f(x) = 2x - 3$; find $f(1)$, $f(-2)$, $f(-1)$

Section 2.2: More on Functions and Their Graphs

Intervals of Increase, Decrease, and Constancy

A function is increasing on interval $I$ if $f(x_1) < f(x_2)$ for $x_1 < x_2$ in $I$; decreasing if $f(x_1) > f(x_2)$; constant if $f(x_1) = f(x_2)$.

• Example: Use graph to identify intervals.

Relative Maxima and Minima

A relative maximum is the highest point in a neighborhood; a relative minimum is the lowest.

• Example: Find relative maxima/minima from graph.

Even and Odd Functions

• Even function: $f(-x) = f(x)$; symmetric about y-axis.

• Odd function: $f(-x) = -f(x)$; symmetric about origin.

Piecewise Functions

Defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} 2x+4 & x < 2 \\ x^2 & x \geq 2 \end{cases}$

Difference Quotient

The difference quotient for $f$ is $\frac{f(x+h)-f(x)}{h}$, $h \neq 0$.

• Example: $f(x) = 6x+1$; find $\frac{f(x+h)-f(x)}{h}$

Section 2.3: Linear Functions and Slopes

Slope of a Line

The slope $m$ is $m = \frac{y_2-y_1}{x_2-x_1}$.

• Example: Find slope through $(-4,5)$ and $(8,7)$.

Point-Slope Form

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

• Example: Slope 4, passes through $(4,3)$: $y-3=4(x-4)$

Slope-Intercept Form

Equation: $y=mx+b$

• Example: Slope 8, passes through $(-4,1)$: $y=8x+b$

Horizontal and Vertical Lines

• Horizontal line: $y=c$; slope is 0.

• Vertical line: $x=c$; slope is undefined.

*Additional info: This guide covers the foundational topics in College Algebra, including algebraic expressions, real numbers, absolute value, polynomials, linear equations, inequalities, functions, graphs, and slopes. All examples and tables are reconstructed and expanded for clarity and completeness.*