College Algebra: Core Concepts, Functions, and Graphs – Study Guide

Quadratic Equations and Their Properties

General Form and Solution Sets

Quadratic equations are polynomial equations of degree two, typically written in the general form:

• General Form: $ax^2 + bx + c = 0$, where $a \neq 0$

• Solution Set: If the solutions (roots) are $r_1$ and $r_2$, the equation can be written as $a(x - r_1)(x - r_2) = 0$

Example: For solution set {2, -1}, the quadratic is $(x - 2)(x + 1) = x^2 - x - 2$.

Inequalities and Interval Notation

Solving Linear Inequalities

To solve inequalities, isolate the variable and express the solution in interval notation.

• Interval Notation: Uses parentheses for open intervals and brackets for closed intervals, e.g., $(-\infty, 2]$

• Example: Solve $(9x + 2)/(x - 5) \leq 3x + 2$

Functions, Relations, Domain, and Range

Definition of a Function

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

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

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

Example: The relation { (1, 11), (1, 22), (3, 8), (3, 10) } is not a function because the input 1 maps to two different outputs (11 and 22).

Tables and Functions

Given a table of x and y values, check if each x maps to only one y to determine if it is a function.

x	y
-5	5
-9	9
-17	17

Domain: {-5, -9, -17} Range: {5, 9, 17}

Linear Equations and Forms

Point-Slope and Slope-Intercept Forms

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

• Slope-Intercept Form: $y = mx + b$

Example: Slope = -2, passing through $(-3, -1/3)$:

• Point-slope: $y + 1/3 = -2(x + 3)$

• Slope-intercept: $y = -2x - 19/3$

Circles: Standard Form

Equation of a Circle

The standard form for a circle with center $(h, k)$ and radius $r$ is:

• $(x - h)^2 + (y - k)^2 = r^2$

Example: Center $(-8, -16)$, $r = \sqrt{8}$: $(x + 8)^2 + (y + 16)^2 = 8$

Transformations of Functions

Square Root and Absolute Value Functions

• Parent Function: $f(x) = \sqrt{x}$ or $f(x) = |x|$

• Transformations: Shifts, stretches, and reflections

Example: $h(x) = 3\sqrt{x + 4}$ is a vertical stretch by 3 and a left shift by 4 units.

Example: $g(x) = |5x + 5| + 3/2$ is a horizontal shift left by 1, vertical stretch by 5, and up by $3/2$.

Function Operations and Composition

Function Addition, Subtraction, Multiplication, and Composition

• Sum: $(f + g)(x) = f(x) + g(x)$

• Difference: $(f - g)(x) = f(x) - g(x)$

• Product: $(fg)(x) = f(x) \cdot g(x)$

• Composition: $(f \circ g)(x) = f(g(x))$

Example: $f(x) = 71x - 2$, $g(x) = x + 10$; $fg(x) = 71x^2 + 712x - 20$

Vertex Form of a Parabola

Vertex Form and Properties

• Vertex Form: $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex

• Shape: Determined by $a$; if $a > 0$, opens upward; if $a < 0$, opens downward

Example: Vertex $(-1, -1)$, $a = 13$: $y = 13(x + 1)^2 - 1$

Polynomial Functions

Graphing and Zeros

• Standard Form: $f(x) = a_nx^n + \ldots + a_0$

• Zeros: Values of $x$ where $f(x) = 0$

• End Behavior: Determined by degree and leading coefficient

Example: $f(x) = (x - 17)(x - 29)$ is a quadratic with zeros at $x = 17$ and $x = 29$.

Example: If $f(x) = x^3 + 13x^2 - 14$ and $x = 1$ is a zero, use synthetic division or factoring to find other zeros.

Polynomial Division

Long Division of Polynomials

• Divide the dividend by the divisor, subtract, bring down the next term, and repeat.

• The result is a quotient and a remainder: $\text{Dividend} = (\text{Divisor}) \times (\text{Quotient}) + \text{Remainder}$

Example: Divide $4x^5 + 2x^3 + 8x^2$ by $5x^2 + 3$.

End Behavior of Polynomial Functions

Leading Coefficient Test

• If degree is even and leading coefficient is positive, both ends rise.

• If degree is even and leading coefficient is negative, both ends fall.

• If degree is odd and leading coefficient is positive, left falls, right rises.

• If degree is odd and leading coefficient is negative, left rises, right falls.

Example: $f(x) = 2x^4 + 19x^2 - 3x - 2$ (even degree, positive leading coefficient): Both ends rise.

Rational Functions and Their Graphs

Key Features: Asymptotes and Intercepts

• Vertical Asymptotes: Values of $x$ that make the denominator zero

• Horizontal Asymptotes: Determined by the degrees of numerator and denominator

• x-intercepts: Values of $x$ that make the numerator zero

Example: $f(x) = \frac{7x^2 - 6x - 16}{4x^2 - 5x}$

Sample Table: Characteristics of a Rational Function

Feature	Value
x-intercepts	(0, 0)
Vertical Asymptotes	x = -2, x = 2
Horizontal Asymptote	y = 0

End Behavior and Asymptotes

• As $x$ approaches a vertical asymptote, $f(x)$ approaches $\infty$ or $-\infty$.

• As $x$ approaches $\pm\infty$, $f(x)$ approaches the horizontal asymptote.

Example: For $f(x)$ with vertical asymptotes at $x = 6$ and $x = 14$, and horizontal asymptote $y = 0$, as $x \to 6^-$, $f(x) \to \infty$ or $-\infty$ depending on the function's sign near the asymptote.

Summary Table: Key Algebraic Forms

Form	General Equation	Key Features
Quadratic	$ax^2 + bx + c$	Parabola, vertex, axis of symmetry
Circle	$(x - h)^2 + (y - k)^2 = r^2$	Center $(h, k)$, radius $r$
Linear	$y = mx + b$	Slope $m$, y-intercept $b$
Rational	$\frac{P(x)}{Q(x)}$	Asymptotes, intercepts

Additional info: This guide covers core College Algebra topics including equations, functions, graphing, transformations, and polynomial/rational analysis, as reflected in the provided questions.