College Algebra Final Exam Review: Comprehensive Study Notes

Equations and Inequalities

Solving Linear and Quadratic Equations

Equations are mathematical statements that assert the equality of two expressions. Solving equations is a fundamental skill in algebra, involving techniques such as factoring, using the quadratic formula, and isolating variables.

• 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:

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

• Example: Solve $x^2 + 4x + 20 = 0$ using the quadratic formula.

• Absolute Value Equations: Equations involving $|x|$ require considering both positive and negative cases.

Solving Inequalities

Inequalities express a range of possible values for a variable. Techniques include isolating the variable and considering the direction of the inequality when multiplying or dividing by negative numbers.

• Linear Inequalities: $ax + b < c$

• Compound Inequalities: $a < x < b$

• Quadratic Inequalities: $ax^2 + bx + c < 0$; solve by finding zeros and testing intervals.

• Example: Solve $-8x + 9 < 6$.

Functions and Graphs

Function Basics

A function is a relation that assigns each input exactly one output. The domain is the set of all possible inputs, and the range is the set of all possible outputs.

• Notation: $f(x)$

• Domain and Range: For $f(x) = |x+1| - 2$, $D: \mathbb{R}$, $R: [-2, \infty)$

• Piecewise Functions: Functions defined by different expressions over different intervals.

Graphing Functions

Graphing involves plotting points and understanding transformations such as shifts, reflections, and stretches.

• Vertex Form: $f(x) = a(x-h)^2 + k$

• Intercepts: Points where the graph crosses the axes.

• Example: Graph $f(x) = |x+1| - 2$ and state domain and range.

Inverse Functions

The inverse of a function reverses the roles of input and output. To find the inverse, solve for $x$ in terms of $y$ and interchange variables.

• Example: Find the inverse of $f(x) = \sqrt{x-3} + 1$.

Polynomial and Rational Functions

Polynomial Functions

Polynomials are expressions of the form $a_nx^n + ... + a_1x + a_0$. Their graphs are smooth and continuous.

• Degree: Highest power of $x$.

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

• End Behavior: Determined by the leading term.

• Example: $f(x) = 3x^2 + 5x$

Rational Functions

Rational functions are ratios of polynomials. Their domain excludes values that make the denominator zero.

• Vertical Asymptotes: Values where the denominator is zero.

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

• Example: $f(x) = \frac{1}{x-3}$

Operations with Functions

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

• Example: $(f+g)(x) = f(x) + g(x)$

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = a^x$. They model growth and decay.

• Compound Interest: $A = P(1 + \frac{r}{n})^{nt}$

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

Logarithmic Functions

Logarithms are the inverses of exponentials. $\log_b(x)$ is the exponent to which $b$ must be raised to get $x$.

• Properties: $\log_b(xy) = \log_b(x) + \log_b(y)$, $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

• Change of Base: $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$

• Example: Solve $\log_x + \log(2x+1) = 1$

Systems of Equations and Inequalities

Solving Systems of Linear Equations

Systems can be solved by substitution, elimination, or using matrices.

• Example: Solve:

  • $x + 4y + z = 2$

  • $7x - 3y = -24$

  • $x - 3y = -3$

• Matrix Methods: Use augmented matrices and row operations.

Matrices and Determinants

Matrix Operations

Matrices are rectangular arrays of numbers. Operations include addition, multiplication, and finding determinants.

• Matrix Multiplication: $(AB)_{ij} = \sum_k A_{ik}B_{kj}$

• Determinant: For $2 \times 2$ matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $\det = ad - bc$

• Example: Find $AB$ for given matrices $A$ and $B$.

Conic Sections

Circles

The equation of a circle with center $(h, k)$ and radius $r$ is:

$(x-h)^2 + (y-k)^2 = r^2$

• Example: Find the equation of a circle with center $(-6,4)$ and radius $3$.

Sequences, Induction, and Probability

Sequences

A sequence is an ordered list of numbers. Arithmetic sequences have a constant difference; geometric sequences have a constant ratio.

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

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

• Example: Find the first five terms of $a_n = 2n - 1$.

Summation Notation

Summation notation $\sum$ is used to represent the sum of a sequence.

• Example: $\sum_{k=1}^5 k^2$

Additional Topics

Domain of Functions

The domain is the set of all input values for which the function is defined. For rational functions, exclude values that make the denominator zero; for even roots, exclude values that make the radicand negative.

• Example: Find the domain of $f(x) = \sqrt{6-x}$

Even and Odd Functions

A function is even if $f(-x) = f(x)$ for all $x$ in the domain, and odd if $f(-x) = -f(x)$.

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

Application Problems

Algebra is used to solve real-world problems such as finding the height of a building using the Pythagorean theorem, or calculating compound interest.

• Example: The base of a 12 ft ladder is 3 ft from a building. Find the height reached using $a^2 + b^2 = c^2$.

Tables

Polynomial Zeros and Multiplicity Table

This table summarizes the zeros, their multiplicities, and the behavior of the graph at each zero.

Factor	Zero	Multiplicity	Effect (touch/cross)
3x	0	1	cross
x-2	2	2	touch
x+1	-1	1	cross

Additional info: Table inferred from the notes for $f(x) = 3x(x-2)^2(x+1)$.

Summary

These notes cover essential topics in College Algebra, including equations, inequalities, functions, graphing, polynomials, rational and exponential functions, logarithms, systems of equations, matrices, conic sections, sequences, and applications. Mastery of these concepts is crucial for success in college-level mathematics.