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

• Linear Equations: Equations of the form $ax + b = 0$.

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

• Solving Techniques: Factoring, completing the square, and the quadratic formula.

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

• Example: Solve $2x^2 + 8x - 10 = 0$ using the quadratic formula.

Inequalities

Inequalities compare two expressions using symbols such as <, >, ≤, or ≥. Solutions are often represented as intervals.

• Example: Solve $3x - 5 < 7$.

• Key Point: When multiplying or dividing both sides by a negative number, reverse the inequality sign.

Functions and Graphs

Definition and Notation

A function is a relation in which each input (domain) has exactly one output (range). Functions are often written as $f(x)$.

• Domain: The set of all possible input values ($x$) for which the function is defined.

• Range: The set of all possible output values ($f(x)$).

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

Graphing Functions

• Intercepts: Points where the graph crosses the axes.

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

• Increasing/Decreasing Intervals: Where the function rises or falls as $x$ increases.

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

• Example: For $f(x)$ graphed, identify intervals where $f(x)$ is increasing or decreasing.

Transformations of Functions

• Vertical and Horizontal Shifts: $f(x) + k$ shifts up/down; $f(x - h)$ shifts right/left.

• Reflections: $-f(x)$ reflects over the $x$-axis; $f(-x)$ reflects over the $y$-axis.

• Stretching/Shrinking: $af(x)$ stretches vertically if $|a| > 1$, shrinks if $0 < |a| < 1$.

• Example: $f(x) = |x|$ reflected over the $x$-axis is $f(x) = -|x|$.

Polynomial and Rational Functions

Polynomial Functions

Polynomial functions are sums of terms of the form $a_nx^n + a_{n-1}x^{n-1} + ... + a_0$ where $n$ is a non-negative integer.

• Degree: The highest power of $x$.

• Zeros/Roots: Values of $x$ where the polynomial equals zero.

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

• Rational Root Theorem: Possible rational roots are $\pm\frac{p}{q}$, where $p$ divides the constant term and $q$ divides the leading coefficient.

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

• Example: Use synthetic division to divide $f(x) = 2x^4 - 5x^3 - 2x - 10$ by $x - 2$.

Rational Functions

Rational functions are ratios of polynomials, $f(x) = \frac{P(x)}{Q(x)}$.

• Domain: All real numbers except where $Q(x) = 0$.

• Vertical Asymptotes: Values of $x$ where $Q(x) = 0$ and $P(x) \neq 0$.

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

• Example: For $f(x) = \frac{2x+1}{x-3}$, $x = 3$ is a vertical asymptote.

Exponential and Logarithmic Functions

Exponential Functions

Exponential functions have the form $f(x) = a \cdot b^x$, where $a \neq 0$ and $b > 0$, $b \neq 1$.

• Growth/Decay: If $b > 1$, the function grows; if $0 < b < 1$, it decays.

• Applications: Compound interest, population growth, radioactive decay.

• Example: $A = Pe^{rt}$ models continuous compound interest.

Logarithmic Functions

Logarithms are the inverses of exponential functions. $y = \log_b x$ 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: $\log_b a = \frac{\log_c a}{\log_c b}$

• Example: Solve $4^{x+1} = 64$ by taking logarithms.

Systems of Equations and Matrices

Solving Systems of Equations

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

• Methods: Substitution, elimination, and matrix methods (Gaussian and Gauss-Jordan elimination).

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

Matrices and Determinants

A matrix is a rectangular array of numbers. Determinants are special numbers calculated from square matrices, useful for solving systems.

• Matrix Operations: Addition, multiplication, finding inverses.

• Determinant of a 2x2 Matrix: For $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, determinant is $ad - bc$.

• Example: Find the determinant of $\begin{bmatrix} 3 & 2 \\ 1 & 4 \end{bmatrix}$.

Sequences, Series, and Probability

Sequences and Series

A sequence is an ordered list of numbers. A series is the sum of the terms of a sequence.

• 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 $a_n = 2n - 5$.

Summation Notation

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

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

Probability

Probability measures the likelihood of an event occurring, expressed as a number between 0 and 1.

• Example: If a coin is flipped, the probability of heads is $\frac{1}{2}$.

Conic Sections

Types of Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone: circles, ellipses, parabolas, and hyperbolas.

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

• Example: Identify the conic section represented by $x^2 - 4y^2 = 16$.

Additional Topics

Even and Odd Functions

• Even Function: $f(-x) = f(x)$ for all $x$ in the domain (symmetric about the $y$-axis).

• Odd Function: $f(-x) = -f(x)$ for all $x$ in the domain (symmetric about the origin).

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

Piecewise-Defined Functions

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

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

Logarithmic and Exponential Equations

• Solving Exponential Equations: Take logarithms of both sides to solve for the variable.

• Solving Logarithmic Equations: Combine logs using properties, then exponentiate both sides.

• Example: Solve $\log(x + 1) - \log(x - 3) = \log(2)$.

Applications

• Compound Interest: $A = P(1 + \frac{r}{n})^{nt}$ or $A = Pe^{rt}$ for continuous compounding.

• Population Growth/Decay: $N(t) = N_0 e^{kt}$

• Mixture Problems: Setting up equations based on totals and concentrations.

Sample Table: Rational Root Theorem Application

Possible Rational Roots	Tested Value	Result
±1, ±2, ±5, ±10	2	f(2) = 0 (Root)
±1, ±2, ±5, ±10	-1	f(-1) ≠ 0
…	…	…

Additional info: Table entries inferred for illustration; actual values depend on the specific polynomial.

Summary of Key Formulas

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

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

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

• Compound Interest: $A = Pe^{rt}$

• Logarithm Properties: $\log_b(xy) = \log_b x + \log_b y$