College Algebra Final Exam Study Guide

Factoring and Quadratic Equations

Factoring Techniques

Factoring is a fundamental skill in algebra used to simplify expressions and solve equations. Common methods include FOIL (First, Outside, Inside, Last) and the box method for multiplying binomials.

• FOIL Method: Used to expand $(a + b)(c + d)$ by multiplying each term.

• Box Method: Organizes multiplication of binomials in a grid for clarity.

• Factoring Quadratics: Express $ax^2 + bx + c$ as $(mx + n)(px + q)$ if possible.

Quadratic Formula

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

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

• Discriminant: $b^2 - 4ac$ determines the nature of the roots (real, repeated, or complex).

Square Root Property

Used when solving equations of the form $x^2 = k$.

• Property: $x^2 = k \Rightarrow x = \pm \sqrt{k}$

Zeros of Polynomial Functions

Finding zeros (roots) of polynomials is essential for graphing and solving equations.

• Factoring: Set each factor equal to zero.

• Long Division: Divides polynomials to simplify or find zeros.

• Synthetic Division: A shortcut for dividing by linear factors.

Linear Equations and Inequalities

Solving Linear Equations

Linear equations are solved for the variable, typically $x$.

• Standard Form: $ax + b = c$

• Solution: Isolate $x$ using inverse operations.

Graphing Inequalities

Inequalities can be graphed on a number line or Cartesian plane.

• Number Line: Use open/closed circles for strict/non-strict inequalities.

• Cartesian Plane: Shade regions representing solutions.

Domain and Range

The domain is the set of possible input values; the range is the set of possible output values.

• Domain: Values $x$ can take.

• Range: Values $y$ can take.

Intercepts

Intercepts are points where the graph crosses the axes.

• Y-intercept: Set $x = 0$.

• X-intercept: Set $y = 0$.

Functions and Their Properties

Function Basics

A function relates each input to exactly one output. Common notation: $f(x)$.

• Domain & Range: As above.

• Graphing: Plot points and analyze shape.

• Example: $f(x) = x^2 + 10$ is a parabola opening upward.

Symmetry Tests

Symmetry helps classify functions and their graphs.

• Even Function: $f(-x) = f(x)$ (symmetric about y-axis).

• Odd Function: $f(-x) = -f(x)$ (symmetric about origin).

Multiplicity of Roots

Multiplicity refers to how many times a root occurs.

• Odd Multiplicity: Graph crosses the axis.

• Even Multiplicity: Graph touches and turns at the axis.

Leading Coefficient and End Behavior

The leading coefficient affects the direction of the graph as $x \to \pm \infty$.

• Positive Leading Coefficient: Parabola opens upward.

• Negative Leading Coefficient: Parabola opens downward.

Transformations

Transformations shift, reflect, or stretch/shrink graphs.

• Shift: Move graph horizontally or vertically.

• Reflect: Flip graph over axis.

• Shrink/Stretch: Compress or expand graph.

Vertex of Quadratics

The vertex is the maximum or minimum point of a parabola.

• Formula: $x = -\frac{b}{2a}$ for $f(x) = ax^2 + bx + c$

Limits and Asymptotes

As $x$ approaches certain values, the function may approach infinity or a constant.

• Vertical Asymptote: Where denominator is zero.

• Horizontal Asymptote: Based on degrees of numerator and denominator.

Exponential and Logarithmic Functions

Exponential and logarithmic functions model growth, decay, and other phenomena.

• Exponential: $y = a e^{kt}$

• Logarithmic: $y = \log_b x$

• Conversion: $y = \log_b x \Leftrightarrow x = b^y$

• Growth & Decay: $A = A_0 e^{kt}$

Piecewise Functions

Piecewise functions have different expressions for different intervals.

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

• Finding Values: Evaluate $f(a)$ based on the interval.

Linear Systems, Matrices, and Determinants

Solving Linear Systems

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

• Substitution: Solve one equation for a variable, substitute into the other.

• Addition (Elimination): Add/subtract equations to eliminate a variable.

• Gaussian Elimination: Systematically reduce to row-echelon form.

Matrix Operations

Matrices are arrays of numbers used to represent systems.

• Addition: Add corresponding elements.

• Multiplication: Multiply rows by columns.

• Scalar Multiplication: Multiply every element by a constant.

• Solving for x: Use inverse matrices or Cramer's Rule.

Determinants and Cramer's Rule

Determinants are used to solve systems and find matrix properties.

• Determinant of 2x2: $\left| \begin{matrix} a & b \\ c & d \end{matrix} \right| = ad - bc$

• Cramer's Rule: $x = \frac{\text{det}(A_x)}{\text{det}(A)}$

Sequences and Series

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers; series are their sums.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• First 4 Terms: Plug $n = 1, 2, 3, 4$ into the formula.

Summation

Summation notation ($\sum$) is used to add terms of a sequence.

• Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Geometric Series: $S_n = a_1 \frac{1 - r^n}{1 - r}$ (for $r \neq 1$)

Finding Specific Terms

To find a specific term, use the sequence formula.

• Example: $a_{28}$ in arithmetic: $a_{28} = a_1 + 27d$

• Example: $a_{28}$ in geometric: $a_{28} = a_1 r^{27}$

Common Ratio and Difference

Identify the pattern in sequences.

• Common Difference (Arithmetic): $d = a_{n} - a_{n-1}$

• Common Ratio (Geometric): $r = \frac{a_{n}}{a_{n-1}}$