College Algebra Final Exam Review: Comprehensive Study Notes

Equations and Inequalities

Solving Linear and Quadratic Equations

Equations are mathematical statements asserting the equality of two expressions. In College Algebra, you will frequently solve linear, quadratic, and higher-degree polynomial equations.

• Linear Equation: An equation of the form ax + b = 0. Solution: $x = -\frac{b}{a}$

• Quadratic Equation: An equation of the form ax^2 + bx + c = 0. Solution: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $2x^2 - 5x + 2 = 0$

Inequalities involve expressions related by <, >, ≤, or ≥. The solution is often an interval or union of intervals.

• Example: Solve $x^2 - 4 < 0$

• Solution: $-2 < x < 2$

Functions and Graphs

Definition and Properties of Functions

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

• Example: $f(x) = x^2 + 1$ has domain $(-\infty, \infty)$ and range $[1, \infty)$

• Zeros of a Function: Values of $x$ for which $f(x) = 0$

Graphing and Transformations

• Vertical Shift: $f(x) + k$ shifts the graph up by $k$ units

• Horizontal Shift: $f(x - h)$ shifts the graph right by $h$ units

• Reflection: $-f(x)$ reflects across the x-axis; $f(-x)$ reflects across the y-axis

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

• Example: The graph of $f(x) = x^2$ shifted right 2 units and up 3 units is $f(x) = (x-2)^2 + 3$

Increasing/Decreasing Intervals and Extrema

• Increasing Interval: Where $f(x)$ rises as $x$ increases

• Decreasing Interval: Where $f(x)$ falls as $x$ increases

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

• Example: $f(x) = -x^2 + 4$ has a maximum at $x = 0$

Polynomial and Rational Functions

Polynomial Functions

• General Form: $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_0$

• Degree: Highest power of $x$

• End Behavior: Determined by the leading term $a_nx^n$

• Zeros and Multiplicity: If $(x - r)^k$ is a factor, $r$ is a zero of multiplicity $k$

• Example: $f(x) = (x-2)^2(x+1)$ has zeros at $x=2$ (multiplicity 2), $x=-1$ (multiplicity 1)

Rational Functions

• Form: $f(x) = \frac{P(x)}{Q(x)}$ where $Q(x) \neq 0$

• Vertical Asymptotes: Values where $Q(x) = 0$

• Horizontal Asymptotes: Determined by degrees of $P(x)$ and $Q(x)$

• Example: $f(x) = \frac{2x}{x-1}$ has a vertical asymptote at $x=1$

Operations with Polynomials

• Synthetic Division: Shortcut for dividing by linear factors

• Remainder Theorem: The remainder of $f(x)$ divided by $x - a$ is $f(a)$

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

Factoring and the Rational Zero Theorem

• Rational Zero Theorem: Possible rational zeros are $\pm \frac{p}{q}$ where $p$ divides the constant and $q$ divides the leading coefficient

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

Exponential and Logarithmic Functions

Exponential Functions

• Form: $f(x) = a \cdot b^x$

• Applications: Population growth, radioactive decay, compound interest

• Example: $A(t) = A_0 e^{kt}$ models exponential growth/decay

Logarithmic Functions

• Definition: $y = \log_b x$ means $b^y = x$

• Properties:

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

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

  • $\log_b(x^k) = k \log_b x$

• Change of Base Formula: $\log_b a = \frac{\log_c a}{\log_c b}$

Solving Exponential and Logarithmic Equations

• Example: Solve $2^x = 8$

• Solution: $x = 3$

• Example: Solve $\log_2 x = 5$

• Solution: $x = 32$

Systems of Equations and Matrices

Solving Systems of Equations

• Substitution and Elimination Methods: Used for solving two or more equations simultaneously

• Gaussian Elimination: Systematically reduces a system to row-echelon form

• Gauss-Jordan Elimination: Reduces to reduced row-echelon form for direct solution

• Example: Solve $\begin{cases} 2x + 3y = 4 \\ x - y = 1 \end{cases}$

Matrices and Determinants

• Matrix Representation: Systems can be written as $AX = B$

• Augmented Matrix: Includes both coefficients and constants

• Determinant: Used to determine invertibility and solve systems

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

Applications

• Word problems involving mixtures, investments, and resource allocation can be modeled with systems of equations.

Sequences, Series, and Mathematical Induction

Sequences

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

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

• Example: Find the first four terms of $a_n = 2n - 6$

Summation Notation

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

• Sum of Geometric Series: $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$

Mathematical Induction

• Principle: Prove a statement for $n=1$, then assume true for $n=k$ and prove for $n=k+1$

Conic Sections

Types of Conic Sections

• Circle: $(x-h)^2 + (y-k)^2 = r^2$

• Parabola: $y = a(x-h)^2 + k$

• Ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

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

Graphing and Identifying Conics

• Recognize conic sections from their equations and graph key features such as vertices, foci, and axes.

Probability and Applications

Basic Probability

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

• Applications: Used in word problems involving counting, arrangements, and selections.

Sample Table: Properties of Polynomial Functions

Degree	End Behavior	Number of Real Zeros	Turning Points
Even	Both ends up or both ends down	0 to n	Up to n-1
Odd	Left and right ends in opposite directions	At least 1	Up to n-1

Additional info:

• Many problems involve interpreting graphs, identifying intervals of increase/decrease, and finding extrema.

• Practice with transformations, symmetry (even/odd functions), and composition of functions is essential.

• Applications include compound interest, population models, and mixture problems.