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. In College Algebra, solving equations involves finding the value(s) of the variable(s) that make the equation true.

• Linear Equation: An equation of the form $ax + b = 0$.

• Quadratic Equation: An equation of the form $ax^2 + bx + c = 0$.

• Example: Solve $2x + 18 = -26$.

  • Subtract 18 from both sides: $2x = -44$

  • Divide by 2: $x = -22$

• Solving Exponential Equations: $4^x = 4^{x+1}$ implies $x = x+1$, which is not possible unless $x$ is infinite. For $6e^x = e^{x+2}$, use logarithms to solve for $x$.

Inequalities

Inequalities compare two expressions and use symbols such as <, >, ≤, or ≥. Solving inequalities involves finding the set of values that satisfy the inequality.

• Example: Solve $x^2 - 2x - 8 > 0$.

• Factor: $(x-4)(x+2) > 0$

• Test intervals: $x < -2$ or $x > 4$

Functions and Graphs

Definition and Properties of Functions

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

• Example: $f(x) = x^2$ has domain $(-, )$ and range $[0, )$.

• One-to-One Function: A function is one-to-one if each output is paired with exactly one input.

Graphing Functions

Graphs visually represent functions. Key features include intercepts, intervals of increase/decrease, and relative extrema.

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

• Relative Maximum/Minimum: Highest/lowest points in a local region of the graph.

• Example: For $f(x) = x^2 - 4$, zeros are $x = 2$ and $x = -2$.

Transformations of Functions

Transformations include shifts, stretches, compressions, and reflections.

• Vertical Shift: $f(x) + k$ shifts the graph up/down.

• Horizontal Shift: $f(x - h)$ shifts the graph right/left.

• Reflection: $-f(x)$ reflects the graph over the x-axis.

Polynomial and Rational Functions

Polynomial Functions

Polynomials are expressions of the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_0$. Their graphs are smooth and continuous.

• Degree: The highest power of $x$.

• Zeros: Solutions to $f(x) = 0$.

• Factoring: Expressing as a product of lower-degree polynomials.

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

Rational Functions

Rational functions are ratios of polynomials. Key features include vertical and horizontal asymptotes.

• Vertical Asymptote: Occurs where the denominator is zero.

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

• Example: $f(x) = \frac{2x}{x^2 - 1}$ has vertical asymptotes at $x = 1$ and $x = -1$.

Synthetic Division and Remainder Theorem

Synthetic division is a shortcut for dividing polynomials by linear factors. The Remainder Theorem states that the remainder of $f(x)$ divided by $x - c$ is $f(c)$.

• Example: Divide $f(x) = 2x^4 - 5x^3 - 2x - 10$ by $x - 1$ using synthetic division.

Exponential and Logarithmic Functions

Exponential Functions

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

• Example: $f(x) = 2^x$

• Applications: Compound interest, population growth.

Logarithmic Functions

Logarithms are the inverses of exponentials. $y = \log_b(x)$ means $b^y = x$.

• Properties:

  • $\log_b(xy) = \log_b(x) + \log_b(y)$

  • $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

  • $\log_b(x^k) = k\log_b(x)$

• Example: Simplify $\log(x^2) - \log(x - 3) = \log(2)$

Systems of Equations and Matrices

Solving Systems of Equations

Systems of equations involve finding values that satisfy multiple equations simultaneously.

• Methods: Substitution, elimination, matrices.

• Example: Solve $2x - 3y = 14$ and $x + 2y = 7$.

Matrices and Determinants

Matrices are rectangular arrays of numbers. Determinants are scalar values that can be computed from square matrices.

• Augmented Matrix: Used to represent systems of equations.

• Gaussian Elimination: A method for solving systems using row operations.

• Example:

  Step	Matrix
  Initial	$\begin{bmatrix} 2 & -3 & | & 14 \\ 1 & 2 & | & 7 \end{bmatrix}$
  Row Operations	Apply to reach row-echelon form.

Sequences, Induction, and Probability

Sequences

A sequence is an ordered list of numbers, often defined by a formula.

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

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

• Example: Write the first four terms of $a_n = 2n - 5$.

Probability

Probability measures the likelihood of an event occurring.

• Formula: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Example: If a bag contains 3 red and 2 blue balls, $P(\text{red}) = \frac{3}{5}$.

Conic Sections

Types of Conic Sections

Conic sections are curves formed by the intersection of a plane and a cone: circles, ellipses, parabolas, and hyperbolas.

• Circle: $x^2 + y^2 = r^2$

• Parabola: $y = ax^2 + bx + c$

• Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

• Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Additional Topics

Function Composition and Inverses

Composition combines two functions: $(f \circ g)(x) = f(g(x))$. The inverse function reverses the effect of the original function.

• Example: If $f(x) = x^2$ and $g(x) = 2x + 1$, then $(f \circ g)(x) = (2x + 1)^2$.

Odd and Even Functions

A function is even if $f(-x) = f(x)$ and odd if $f(-x) = -f(x)$.

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

Domain and Range

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

• Example: For $f(x) = \sqrt{x-2}$, domain is $x \geq 2$.

Asymptotes

Asymptotes are lines that a graph approaches but never touches.

• Vertical Asymptote: Set denominator equal to zero.

• Horizontal Asymptote: Compare degrees of numerator and denominator.

Logarithmic Equations

Logarithmic equations can be solved using properties of logarithms.

• Example: $\log(x^2) - \log(x - 3) = \log(2)$

Applications

College Algebra concepts are applied in finance, science, and everyday problem-solving.

• Compound Interest: $A = Pe^{rt}$

• Population Growth: $N(t) = N_0e^{kt}$

• Mixture Problems: Use systems of equations to solve for unknowns.

Summary Table: Types of Functions and Their Properties

Type	General Form	Key Properties
Linear	$y = mx + b$	Straight line, constant rate of change
Quadratic	$y = ax^2 + bx + c$	Parabola, vertex, axis of symmetry
Polynomial	$y = a_nx^n + ... + a_0$	Degree, zeros, end behavior
Rational	$y = \frac{p(x)}{q(x)}$	Asymptotes, domain restrictions
Exponential	$y = a^x$	Rapid growth/decay, horizontal asymptote
Logarithmic	$y = \log_b(x)$	Inverse of exponential, domain $x > 0$

Practice Problems and Applications

• Find the domain and range of $f(x) = \frac{1}{x-2}$.

• Solve the system: $x + y = 5$, $2x - y = 4$.

• Graph $f(x) = x^3 - 2x^2 - 5x + 6$ and identify zeros and extrema.

• Write the first four terms of the sequence $a_n = 2n - 5$.

• Determine if $f(x) = x^2$ is even, odd, or neither.

Additional info: These notes expand on the exam review questions by providing definitions, examples, and formulas for each major College Algebra topic. The summary table classifies function types and their properties for quick reference.