College Algebra Exam 1 Review: Comprehensive Study Notes

Equations and Factoring

Solving Linear and Quadratic Equations

Solving equations is a foundational skill in algebra, involving finding the value(s) of the variable that make the equation true.

• Linear Equations: Equations of the form $ax + b = 0$ can be solved by isolating $x$.

• Quadratic Equations: Equations of the form $ax^2 + bx + c = 0$ can be solved by factoring, completing the square, or using the quadratic formula:

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

• Factoring: Expressing a polynomial as a product of its factors. For example, $x^2 - 5x + 6 = (x-2)(x-3)$.

• Prime Polynomial: A polynomial that cannot be factored over the integers.

Example: Factor $x^2 - 9$.

• Solution: $x^2 - 9 = (x-3)(x+3)$

Quadratic Equations and Applications

Solving by Factoring and Quadratic Formula

Quadratic equations can be solved by factoring when possible, or by applying the quadratic formula when factoring is not feasible.

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

• Discriminant: $D = b^2 - 4ac$ determines the nature of the roots:

Discriminant ($D$)	Nature of Roots
$D > 0$	Two real and distinct roots
$D = 0$	One real repeated root
$D < 0$	Two complex roots

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

Absolute value equations involve expressions like $|x| = a$. The solution depends on the value of $a$.

• If $|x| = a$ and $a \geq 0$, then $x = a$ or $x = -a$.

• If $|x| = a$ and $a < 0$, there is no solution.

Example: Solve $|x - 3| = 5$.

• Solution: $x - 3 = 5$ or $x - 3 = -5$; $x = 8$ or $x = -2$.

Solving Absolute Value Inequalities

• For $|x| < a$: $-a < x < a$

• For $|x| > a$: $x < -a$ or $x > a$

Functions and Relations

Definition and Identification

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 (usually $x$ values).

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

• Vertical Line Test: A graph represents a function if no vertical line intersects the graph at more than one point.

Example: The set $\{(1,2), (2,3), (3,4)\}$ is a function.

Evaluating Functions

• To evaluate $f(x)$ at $x = a$, substitute $a$ into the function.

Example: If $f(x) = x^2 + 1$, then $f(2) = 2^2 + 1 = 5$.

Graphing and Analyzing Functions

Domain and Range from Graphs

To determine the domain and range from a graph, observe the extent of the graph along the $x$-axis (domain) and $y$-axis (range).

• Use interval notation to express domain and range, e.g., $(-\infty, 3]$.

Intercepts

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

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

Increasing, Decreasing, and Constant Intervals

• A function is increasing on intervals where its graph rises as $x$ increases.

• A function is decreasing on intervals where its graph falls as $x$ increases.

• A function is constant on intervals where its graph is flat.

Relative Maximum and Minimum

• Relative Maximum: A point where the function changes from increasing to decreasing.

• Relative Minimum: A point where the function changes from decreasing to increasing.

Even and Odd Functions; Symmetry

Definitions

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

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

• Neither: If neither condition is met.

Piecewise Functions

Definition and Evaluation

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

• To evaluate, determine which interval the input belongs to and use the corresponding expression.

Example: $f(x) = \begin{cases} x^2 & x < 0 \\ 2x + 1 & x \geq 0 \end{cases}$

Difference Quotient

Definition

The difference quotient is used to compute the average rate of change of a function over an interval:

$\frac{f(x+h) - f(x)}{h}$

• Important in calculus for defining the derivative.

Linear Equations and Slope

Slope and Equation of a Line

• Slope ($m$): Measures the steepness of a line. $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Slope-Intercept Form: $y = mx + b$

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

• Standard Form: $Ax + By = C$

Parallel and Perpendicular Lines

• Parallel Lines: Have equal slopes ($m_1 = m_2$).

• Perpendicular Lines: Slopes are negative reciprocals ($m_1 \cdot m_2 = -1$).

Vertical and Horizontal Lines

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

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

Rates of Change

Average Rate of Change

The average rate of change of a function $f$ from $x = a$ to $x = b$ is:

$\frac{f(b) - f(a)}{b - a}$

• Represents the slope of the secant line through $(a, f(a))$ and $(b, f(b))$.

Summary Table: Key Algebraic Concepts

Concept	Definition	Example
Quadratic Formula	Solves $ax^2 + bx + c = 0$	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Absolute Value Equation	$|x| = a \implies x = a$ or $x = -a$	$|x| = 3 \implies x = 3$ or $x = -3$
Function	Each input has one output	$f(x) = x^2$
Difference Quotient	Average rate of change	$\frac{f(x+h) - f(x)}{h}$
Slope	Steepness of a line	$m = \frac{y_2 - y_1}{x_2 - x_1}$

Additional info: These notes synthesize the main topics and skills reviewed in a College Algebra Exam 1, including solving equations, analyzing functions, graphing, and interpreting algebraic relationships. All formulas and definitions are standard for a first-semester college algebra course.