College Algebra Final Exam Comprehensive Study Guide

Equations and Inequalities

Solving Linear and Quadratic Equations

Equations are mathematical statements that assert the equality of two expressions. In College Algebra, solving equations is a fundamental skill, including linear, quadratic, and rational equations.

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

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

• Rational Equation: An equation involving fractions whose numerators and/or denominators contain a variable.

• Example: Solve $2x - 4 = 10$.

  • $2x = 14$

  • $x = 7$

Inequalities

Inequalities compare two expressions and use symbols such as <, >, ≤, or ≥. Solutions are often expressed as 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 each input exactly one output. The domain is the set of all possible inputs, and the range is the set of all possible outputs.

• Domain: Set of all possible input values (x-values).

• Range: Set of all possible output values (y-values).

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

Graphing Functions

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

• Intercepts: Points where the graph crosses the axes.

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

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

• Example: For $f(x)$ shown in a graph, identify intervals where $f(x)$ is increasing or decreasing.

Transformations of Functions

Transformations shift, stretch, compress, or reflect graphs.

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

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

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

• Example: $g(x) = f(x - 2)$ shifts $f(x)$ right by 2 units.

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 have smooth, continuous curves.

• Degree: Highest power of x.

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

• End Behavior: Determined by leading term.

• Example: $f(x) = x^3 - 4x^2 + x - 6$

Rational Functions

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

• Vertical Asymptote (VA): Values of x where denominator is zero.

• Horizontal Asymptote (HA): Determined by degrees of numerator and denominator.

• Example: $f(x) = \frac{2x}{x^2 - 4}$ has VA at $x = 2, -2$.

Factoring and Zeros

Factoring polynomials helps find zeros and simplify expressions.

• Rational Zero Theorem: Possible rational zeros are factors of constant term over leading coefficient.

• Example: For $f(x) = 2x^3 - 5x^2 - 27x - 10$, possible rational zeros are $\pm1, \pm2, \pm5, \pm10$.

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: Solve $\log_2(x) = 3$; $x = 8$.

Systems of Equations and Matrices

Solving Systems of Equations

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

• Example: Solve $2x + 3y = 7$ and $x - y = 1$.

Matrices and Determinants

Matrices organize data and can be used to solve systems. The determinant helps determine if a system has a unique solution.

• Matrix: Rectangular array of numbers.

• Determinant: Scalar value from a square matrix.

• Example: For $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$, $\det(A) = 2 \times 4 - 1 \times 3 = 5$.

Gaussian and Gauss-Jordan Elimination

These are systematic methods for solving systems using row operations.

• Row Operations: Swap, scale, or add rows to simplify.

• Example: Reduce $\begin{bmatrix} 2 & -4 & 12 \\ 1 & 2 & 4 \end{bmatrix}$ to 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: Find first four terms of $a_n = 2n - 6$.

Series and Summation

The sum of terms in a sequence is called a series.

• Example: $\sum_{n=1}^{10} n = 55$

Probability

Probability measures the likelihood of an event.

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

• Example: Probability of rolling a 3 on a six-sided die: $P = \frac{1}{6}$

Conic Sections

Types of Conic Sections

Conic sections include circles, ellipses, parabolas, and hyperbolas, each defined by a specific equation.

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

Tables

Example: Rational Zero Candidates

The Rational Zero Theorem provides a list of possible rational zeros for a polynomial.

Possible Zeros	Tested Value	Result
1	12	47
2	–	–
–	–	–

Additional info: Table entries inferred from synthetic division and remainder theorem context.

Additional Info

• Odd and even functions: Even functions are symmetric about the y-axis; odd functions are symmetric about the origin.

• One-to-one functions: Pass the horizontal line test.

• Logarithmic properties: Used to simplify and solve equations.

• Applications: Problems include compound interest, population growth, and mixture problems.