College Algebra: Comprehensive Study Guide and Practice Questions

Equations and Solutions

Solving Linear and Quadratic Equations

Equations are mathematical statements that assert the equality of two expressions. In College Algebra, solving equations is a fundamental skill, including linear, quadratic, and radical equations.

• Linear Equations: An equation of the form ax + b = 0. The solution is $x = -\frac{b}{a}$.

• Quadratic Equations: An equation of the form ax^2 + bx + c = 0. Solutions are found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Radical Equations: Equations involving roots, such as $\sqrt{x}$ or $\sqrt{ax + b}$, often require squaring both sides to eliminate the radical.

Example: Solve $2x^2 - 10x = 50$ and find the sum of the solutions.

• Rewrite as $2x^2 - 10x - 50 = 0$

• Apply the quadratic formula:

• $x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 2 \cdot (-50)}}{2 \cdot 2}$

• Sum of solutions for $ax^2 + bx + c = 0$ is $-\frac{b}{a}$

Functions and Their Properties

Function Notation and Evaluation

A function is a relation that assigns each input exactly one output. Function notation is written as $f(x)$, where $x$ is the input variable.

• Evaluating Functions: Substitute the given value into the function.

• Example: If $f(x) = 2x^2 + x + 1$, then $f(2) = 2(2)^2 + 2 + 1 = 8 + 2 + 1 = 11$.

Domain of a Function

The domain of a function is the set of all possible input values (typically $x$) for which the function is defined.

• For rational functions $f(x) = \frac{1}{x-3}$, the domain excludes $x = 3$.

• For square root functions $f(x) = \sqrt{x-3}$, the domain is $x \geq 3$.

Odd and Even Functions

Odd functions satisfy $f(-x) = -f(x)$ for all $x$ in the domain. Even functions satisfy $f(-x) = f(x)$.

• Example: $f(x) = x^3$ is odd; $f(x) = x^2$ is even.

Algebraic Manipulation

Simplifying Expressions

Simplifying algebraic expressions involves combining like terms, factoring, and rationalizing denominators.

• Example: Simplify $\frac{6}{a} + \frac{8}{b}$ as $\frac{6b + 8a}{ab}$.

Factoring

Factoring is expressing an expression as a product of its factors.

• Example: $x^2 - 5x + 6 = (x - 2)(x - 3)$

Systems of Equations

Solving Systems

A system of equations consists of two or more equations with the same variables. Solutions are values that satisfy all equations simultaneously.

• Methods: Substitution, elimination, and graphical methods.

• Example: Solve $y = \frac{1}{2}x$, $y = x^2 + 1$, $x - y = 1$.

Graphing and Transformations

Graphing Linear Equations

The graph of a linear equation $y = mx + b$ is a straight line with slope $m$ and y-intercept $b$.

• Finding the Equation of a Line: Use the point-slope form $y - y_1 = m(x - x_1)$.

• Parallel Lines: Have the same slope.

Transformations of Functions

Transformations include shifting, stretching, and compressing graphs.

• Vertical Shifts: $f(x) + k$ shifts up by $k$ units.

• Horizontal Shifts: $f(x - h)$ shifts right by $h$ units.

• Stretch/Compression: $af(x)$ stretches vertically by factor $a$.

Inverse Functions

Finding the Inverse

The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$. To find the inverse, solve $y = f(x)$ for $x$ in terms of $y$.

• Example: For $f(x) = 2x - 3$, set $y = 2x - 3$, solve for $x$:

• $y = 2x - 3 \implies x = \frac{y + 3}{2}$

• So, $f^{-1}(x) = \frac{x + 3}{2}$

Summary Table: Function Transformations

Transformation	Effect on Graph	Example
Vertical Shift	Up/down by $k$ units	$f(x) + k$
Horizontal Shift	Right/left by $h$ units	$f(x - h)$
Vertical Stretch/Compression	Stretched/compressed by factor $a$	$af(x)$
Reflection	Across x-axis or y-axis	$-f(x)$ or $f(-x)$

Additional info:

• Some questions involve finding the sum of solutions, which for quadratics is $-\frac{b}{a}$.

• Questions on function composition: $f(g(x))$ means substitute $g(x)$ into $f$.

• Domain questions require identifying values that make denominators zero or radicands negative.