College Algebra Final Exam Study Guide: Key Concepts and Methods

Rational Equations

Solving Rational Equations

Rational equations involve fractions with polynomials in the numerator and denominator. Solving these requires careful handling of denominators and checking for extraneous solutions.

• Step 1: Find the Least Common Denominator (LCD) of all terms.

• Step 2: Multiply each term by the LCD and simplify.

• Step 3: Solve for x.

• Step 4: Check solutions against exclusions (values that make any denominator zero).

Exclusions: Set each denominator not equal to zero and solve for excluded values.

Example: Solve $\frac{1}{x} + \frac{2}{x+1} = 3$

Find LCD, multiply through, solve, and check for excluded values.

Quadratic Equations

Quadratic Formula

The quadratic formula solves equations of the form $ax^2 + bx + c = 0$.

• Standard Form: $ax^2 + bx + c = 0$

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

Example: Solve $2x^2 + 3x - 2 = 0$ using the formula.

Factoring Quadratics

Factoring is another method to solve quadratic equations.

• Step 1: Find the Greatest Common Factor (GCF).

• Step 2: Factor the quadratic (may require multiple steps).

• Step 3: Set each factor equal to zero and solve for x.

• Step 4: Check solutions.

Radical Equations

Solving Radical Equations

Radical equations contain variables under a root. Isolating the radical and squaring both sides is key.

• Step 1: Isolate the radical.

• Step 2: Square both sides.

• Step 3: Solve for x.

• Step 4: Check for extraneous solutions.

Absolute Value Equations and Inequalities

Solving Absolute Value Equations

Absolute value equations require splitting into two cases.

• Step 1: Isolate the absolute value expression.

• Step 2: Write two equations: one with positive, one with negative right side.

• Step 3: Solve for x.

• Step 4: Interpret solutions based on the value of c (positive: two solutions, zero: one solution, negative: no solution).

Solving Absolute Value Inequalities

Absolute value inequalities are handled differently based on the inequality sign.

• For ≥ or >: Write two inequalities using ±c.

• For ≤ or <: Write a compound inequality using ±c.

Solving Inequalities

Linear Inequalities

Solving linear inequalities involves similar steps to equations, but solutions are often written in interval notation.

• Step 1: Combine like terms and isolate x on the left, constants on the right.

• Step 2: Solve for x.

• Step 3: Express the solution in interval notation.

Functions

Definition and Types

A function is a relation where each input x corresponds to exactly one output y.

• One-to-One Function: Each y value is paired with only one x.

• Piecewise Function: Defined by different expressions depending on the value of x.

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

Finding Domain

The domain is the set of all possible input values for a function.

• Step 1: Assume all real numbers ($\mathbb{R}$).

• Step 2: Identify restrictions:

  • x in denominator: Set denominator ≠ 0 and solve.

  • x under even root: Set radicand ≥ 0 and solve.

• Step 3: Write domain in interval notation.

Algebra of Functions

Operations and Compositions

Functions can be added, subtracted, multiplied, divided, and composed.

• Step 1: Write operation as definition.

• Step 2: Substitute values.

• Step 3: Distribute and combine like terms.

• Step 4: Solve as needed.

Composite Functions

A composite function is formed by applying one function to the result of another: $f(g(x))$.

• Step 1: Write as definition.

• Step 2: Substitute values.

• Step 3: Simplify.

Decomposing Composite Functions

Decomposition involves identifying the inner and outer functions.

• Step 1: Identify $g(x)$ (the more complex part).

• Step 2: Identify $f(x)$ (replace $g(x)$ with $x$).

Inverse Functions

The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$.

• Step 1: Replace $f(x)$ with $y$.

• Step 2: Switch $x$ and $y$.

• Step 3: Solve for $y$.

• Step 4: Replace $y$ with $f^{-1}(x)$.

Quadratic Functions and Parabolas

Vertex and Axis of Symmetry

The vertex of a parabola is its maximum or minimum point. The axis of symmetry passes through the vertex.

• Vertex: $(h, k)$ where $h = \frac{-b}{2a}$ and $k = f(h)$

• Axis of Symmetry: $x = h$

Example: For $f(x) = 2x^2 - 4x + 1$, $h = \frac{4}{4} = 1$, $k = f(1)$

Graphing Parabolas

• a > 0: Parabola opens upward (smiley face).

• a < 0: Parabola opens downward (frowny face).

Systems of Equations and Determinants

Solving Systems of Linear Equations

Systems can be solved by substitution, addition (elimination), or matrix methods.

• Substitution: Use when a variable has no coefficient.

• Addition: Use when you can create opposites.

• Step 1: Solve for one variable.

• Step 2: Substitute into the other equation.

• Step 3: Solve for the second variable.

• Step 4: Write as an ordered pair and check.

Determinants and Cramer's Rule

Determinants are used to solve systems using matrices.

• First and Second Order Determinant: $D = \left| \begin{array}{cc} a_1 & b_1 \\ a_2 & b_2 \end{array} \right| = a_1b_2 - a_2b_1$

• Cramer's Rule:

Determinant	Formula
$D$	$\left| \begin{array}{cc} a_1 & b_1 \\ a_2 & b_2 \end{array} \right|$
$D_x$	$\left| \begin{array}{cc} c_1 & b_1 \\ c_2 & b_2 \end{array} \right|$
$D_y$	$\left| \begin{array}{cc} a_1 & c_1 \\ a_2 & c_2 \end{array} \right|$
$x$	$\frac{D_x}{D}$
$y$	$\frac{D_y}{D}$

Third-Order Determinant:

$\left| \begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array} \right| = a_1 \left| \begin{array}{cc} b_2 & c_2 \\ b_3 & c_3 \end{array} \right| - a_2 \left| \begin{array}{cc} b_1 & c_1 \\ b_3 & c_3 \end{array} \right| + a_3 \left| \begin{array}{cc} b_1 & c_1 \\ b_2 & c_2 \end{array} \right|$

Exponential and Logarithmic Functions

Logarithmic and Exponential Forms

Logarithms and exponentials are inverse operations.

• Logarithmic Form: $\log_b x = y$

• Exponential Form: $b^y = x$

Change of Base Formula: $\log_b M = \frac{\log M}{\log b}$

Properties of Logarithms

• Multiplication: $\log_b (MN) = \log_b M + \log_b N$

• Division: $\log_b \left(\frac{M}{N}\right) = \log_b M - \log_b N$

• Powers: $\log_b (M^p) = p \log_b M$

Order for Expansion: 1. Quotients/Products 2. Powers

Order for Condensing: 1. Powers 2. Products/Quotients

Solving Logarithmic Equations

• Step 1: Condense to one log = one log.

• Step 2: Set arguments equal: $\log_a M = \log_a N \implies M = N$

• Step 3: Solve for x.

• Step 4: Check that arguments are positive.

Solving Exponential Equations

• Step 1: Isolate exponential expression.

• Step 2: Take log or ln of both sides.

• Step 3: Use power rule to simplify.

• Step 4: Solve for x.

• Step 5: Check solution.

Example: $2^x = 8 \implies x = 3$

Compound Interest

Formulas

Compound interest is calculated using exponential functions.

• General Formula: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• Continuous Compounding: $A = Pe^{rt}$

Example: $P = 1000$, $r = 0.05$, $n = 4$, $t = 2$

Systems of Linear Inequalities

Graphing Linear Inequalities

Graphing linear inequalities involves shading regions that satisfy the inequality.

• Step 1: Replace inequality with equality and graph the line.

• Step 2: Use solid line for ≤ or ≥, dashed for < or >.

• Step 3: Choose a test point not on the line.

• Step 4: Substitute test point; if true, shade that region.

Solving by Addition (Elimination)

• Step 1: Create opposites for variables.

• Step 2: Add equations to eliminate one variable.

• Step 3: Solve for remaining variable.

• Step 4: Substitute back and solve for other variable.

Graphing Multiple Inequalities

Graph each inequality separately, then shade the region that satisfies all.

Matrix Methods

Gauss-Jordan Elimination

Gauss-Jordan elimination is a systematic method for solving systems using matrices.

• Step 1: Build the augmented matrix.

• Step 2: Use row operations to reduce to row-echelon form.

• Step 3: Solve for variables.

Cramer's Rule (2x2 and 3x3)

See determinants section above for formulas.

• Step 1: Build matrix.

• Step 2: Find determinant $D$.

• Step 3: Find $D_x$ and $D_y$.

• Step 4: Compute $x = \frac{D_x}{D}$, $y = \frac{D_y}{D}$.

• Step 5: Write as ordered pair and check solution.

Third-Order Determinant

For a 3x3 matrix:

$\left| \begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{array} \right| = a_1 \left| \begin{array}{cc} b_2 & c_2 \\ b_3 & c_3 \end{array} \right| - a_2 \left| \begin{array}{cc} b_1 & c_1 \\ b_3 & c_3 \end{array} \right| + a_3 \left| \begin{array}{cc} b_1 & c_1 \\ b_2 & c_2 \end{array} \right|$

Additional info: These notes cover all major topics in college algebra, including equations, inequalities, functions, systems, matrices, determinants, and exponential/logarithmic functions. For exam preparation, practice solving each type of equation and graphing functions and inequalities.