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

Section P.1: Algebraic Expressions and Real Numbers

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. Variables are letters used to represent unknown values.

• Variable: A symbol (often a letter) representing an unknown value.

• Algebraic Expression: A combination of variables, numbers, and operations (addition, subtraction, multiplication, division, exponents, 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(s) for the variable(s) and simplify using the order of operations.

• Order of Operations:

  1. Parentheses and other grouping symbols

  2. Exponents and roots

  3. Multiplication and division (left to right)

  4. Addition and 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 $\frac{4(x-6)}{2x+1}$ for $x = 8$.

Ordering Real Numbers and Absolute Value

Real numbers can be represented on a number line. Inequality symbols are used to compare their order.

• Number Line: Numbers increase from left to right.

• Inequality Symbols: $<$ (less than), $>$ (greater than), $\leq$ (less than or equal to), $\geq$ (greater than or equal to).

• Absolute Value: The distance from 0 on the number line, always nonnegative. $|a| = a$ if $a \geq 0$, $|a| = -a$ if $a < 0$.

Example: $|-5| = 5$

Example: $|x|$ for $x = 6$

Section 1.1: Graphs and the Rectangular Coordinate System

Plotting Points

The rectangular coordinate system (Cartesian plane) consists of two perpendicular axes: the x-axis (horizontal) and y-axis (vertical). Each point is represented as an ordered pair $(x, y)$.

• Quadrants: The plane is divided into four quadrants.

• Origin: The point $(0, 0)$ where the axes intersect.

Example: Plot the points $A(2,3)$, $B(-5,7)$, $C(0,-8)$, $D(4,-6)$, $E(-4,0)$, $F(-3,-5)$.

Intercepts

• x-intercept: The point where a graph crosses the x-axis ($y=0$).

• y-intercept: The point where a graph crosses the y-axis ($x=0$).

Graphing Absolute Value Functions

The basic absolute value function is $f(x) = |x|$. Its graph is a 'V' shape, symmetric about the y-axis.

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

Example Table:

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 of Polynomials

Polynomials are added or subtracted by combining like terms (terms with the same variable and exponent).

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

Multiplication of Polynomials

• Product of Monomials: Multiply coefficients and add exponents: $a^m \cdot a^n = a^{m+n}$

• Distributive Property: $a(b + c) = ab + ac$

• FOIL Method: For $(a+b)(c+d)$, multiply First, Outside, Inside, Last terms.

Example: $(5x-1)(2x+3)$

Section 1.2: Solving Linear Equations

Linear Equations in One Variable

A linear equation can be written in the form $ax + b = 0$, where $a$ and $b$ are real numbers and $a \neq 0$.

• Steps:

  1. Simplify both sides (distribute, combine like terms)

  2. Collect variable terms on one side

  3. Add/subtract to isolate the variable

  4. Multiply/divide to solve for the variable

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

Section 1.7: Interval Notation and Linear Inequalities

Interval Notation

Interval notation is used to represent sets of real numbers on the number line.

Interval Notation	Set-Builder Notation	Description
(a, b)	{x | a < x < b}	All x between a and b, not including a or b
[a, b]	{x | a ≤ x ≤ b}	All x between a and b, including a and b
(a, b]	{x | a < x ≤ b}	All x between a and b, including b
[a, b)	{x | a ≤ x < b}	All x between a and b, including a
(a, ∞)	{x | x > a}	All x greater than a
[a, ∞)	{x | x ≥ a}	All x greater than or equal to a
(-∞, b)	{x | x < b}	All x less than b
(-∞, b]	{x | x ≤ b}	All x less than or equal to b

Solving Linear Inequalities

• Additive Property: You can add or subtract the same value from both sides.

• Multiplicative Property: You can multiply or divide both sides by a positive number. If you multiply or divide by a negative number, reverse the inequality sign.

Example: $3x - 5 \geq 4$

Section 2.1: Basics of Functions and Their Graphs

Domain and Range

The domain of a relation is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

Example: For the relation {(0,5), (1,-2), (4,2), (3,9)}, the domain is {0,1,3,4}, the range is {5,-2,2,9}.

Functions

A function is a relation in which each input (domain) is paired with exactly one output (range).

• Vertical Line Test: If any vertical line crosses a graph more than once, it is not a function.

Function Notation and Evaluation

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

Example: If $f(x) = x^2 - 2x - 5$, find $f(-1)$, $f(0)$, $f(2)$, $f(-3)$.

Obtaining Information from Graphs

• Domain: Input values (x-axis)

• Range: Output values (y-axis)

• x-intercept: Where the graph crosses the x-axis ($y=0$)

• y-intercept: Where the graph crosses the y-axis ($x=0$)

Section 2.2: More on Functions and Their Graphs

Intervals of Increase, Decrease, and Constancy

• Increasing: $f(x)$ increases as $x$ increases on an interval.

• Decreasing: $f(x)$ decreases as $x$ increases on an interval.

• Constant: $f(x)$ remains the same as $x$ increases on an interval.

Relative Maxima and Minima

• Relative Maximum: The highest point in a local region of the graph.

• Relative Minimum: The lowest point in a local region of the graph.

Even and Odd Functions

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain; symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain; symmetric about the origin.

Piecewise Functions

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

Example: $f(x) = \begin{cases} 2x+5 & x < 4 \\ x-1 & x \geq 4 \end{cases}$

Difference Quotient

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

Example: For $f(x) = 6x + 1$, find and simplify the difference quotient.

Section 2.3: Linear Functions and Slopes

Slope of a Line

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

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

Example: Find the slope between $(-4,5)$ and $(8,7)$.

Equations of Lines

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

• Slope-Intercept Form: $y = mx + b$

Example: Write the equation of a line with slope 4 passing through $(4,3)$.

Graphing Horizontal and Vertical Lines

• Horizontal Line: $y = b$ (slope = 0)

• Vertical Line: $x = a$ (slope undefined)

Example: Graph $y = 3$ and $x = 1$.

Additional info: These notes cover foundational topics in College Algebra, including algebraic expressions, real numbers, polynomials, linear equations, inequalities, functions, and graphing. The content is structured to support exam preparation and conceptual understanding.