Comprehensive Precalculus Final Review: Key Concepts and Practice Problems

Algebraic Expressions and Factoring

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Simplifying involves factoring and reducing common terms.

• Key Steps: Factor both numerator and denominator completely.

• Cancel any common factors.

• State restrictions on the variable (values that make the denominator zero).

• Example: Simplify $\frac{16x^2 - 7}{2x - 1}$.

Factoring Polynomials

Factoring is expressing a polynomial as a product of its factors.

• Key Steps: Look for common factors, difference of squares, trinomials, or grouping.

• Example: Completely factor $12x^3 - 27x^2$.

Rational Functions and Asymptotes

Identifying Asymptotes

Rational functions can have vertical, horizontal, or oblique (slant) asymptotes.

• Vertical Asymptotes: Set the denominator equal to zero and solve for $x$.

• Horizontal Asymptotes: Compare the degrees of numerator and denominator.

• Oblique Asymptotes: Occur when the degree of the numerator is one more than the denominator.

• Example: For $f(x) = \frac{2x^2 + 3x - 2}{x^2 - 1}$, find all asymptotes.

Functions and Their Properties

Evaluating and Graphing Functions

Functions can be represented algebraically, graphically, or verbally. Understanding their domain, range, and behavior is essential.

• Domain: All possible input values ($x$) for which the function is defined.

• Range: All possible output values ($y$).

• Example: Find the domain of $f(x) = \sqrt{8 - x}$.

Piecewise Functions

Piecewise functions are defined by different expressions over different intervals.

• Example: $f(x) = \begin{cases} x^2 + 2x - 5 & x < 0 \\ \sqrt{x + 1} & x \geq 0 \end{cases}$

Equations and Inequalities

Solving Equations

Equations may be linear, quadratic, or involve rational or radical expressions.

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

• Example: Solve $4x^2 - 8x - 3 = 0$.

Solving Inequalities

Inequalities can be solved algebraically or graphically. Express solutions in interval notation.

• Example: Solve $|4x - 7| \leq 5$.

Polynomials and Their Graphs

Zeros and Intercepts

The zeros of a polynomial are the $x$-values where the function equals zero. Intercepts are points where the graph crosses the axes.

• Finding Zeros: Set $f(x) = 0$ and solve for $x$.

• Example: Find all real zeros of $f(x) = x^3 - 2x^2 - 5x + 6$.

End Behavior

The end behavior of a polynomial describes how the function behaves as $x \to \infty$ or $x \to -\infty$.

• Leading Coefficient Test: Determines the direction of the graph's ends.

Exponential and Logarithmic Functions

Properties and Equations

Exponential and logarithmic functions are inverses of each other. They are used to model growth and decay.

• Exponential Form: $y = ab^{x}$

• Logarithmic Form: $y = \log_b x$

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

• Example: Solve $\log_2(x + 3) - \log_2(x - 6) + 7 = \log_5 5$.

Trigonometry

Angles and Right Triangles

Trigonometric functions relate the angles of a triangle to the lengths of its sides.

• Basic Functions: $\sin \theta$, $\cos \theta$, $\tan \theta$

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$

• Example: Find the equation of the line perpendicular to $2x - 5y = 6$ and passing through $(4, 3)$.

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 center and radius of the circle with equation $x^2 + y^2 + 8x + 10y = 7$.

Sequences, Series, and Probability

Arithmetic and Geometric Sequences

Sequences are ordered lists of numbers following a specific pattern.

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

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

• Example: Find the sum of the first 10 terms of an arithmetic sequence with $a_1 = 3$, $d = 2$.

Probability

Probability measures the likelihood of an event occurring.

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

Graphing and Analysis

Graphing Functions

Graphing involves plotting points and analyzing the behavior of functions, including intercepts, asymptotes, and end behavior.

• Example: Graph $f(x) = \frac{1}{x}$ and state the domain and range.

Transformations

Transformations include translations, reflections, stretches, and compressions.

• Example: Describe the effect of $f(x) = (x - 2)^2 + 3$ compared to $f(x) = x^2$.

Applications and Word Problems

Modeling with Functions

Functions can model real-world scenarios such as revenue, cost, and optimization problems.

• Example: The price $p$ (in dollars) and the quantity $x$ of a product sold obey the demand equation $p = 100 - 0.01x$.

• Find the revenue function $R(x)$, maximum revenue, and the quantity that maximizes revenue.

Summary Table: Types of Functions and Their Properties

Type	General Form	Key Features
Linear	$y = mx + b$	Straight line, constant rate of change
Quadratic	$y = ax^2 + bx + c$	Parabola, vertex, axis of symmetry
Polynomial	$y = a_nx^n + \ldots + a_0$	Multiple zeros, end behavior
Rational	$y = \frac{p(x)}{q(x)}$	Asymptotes, holes
Exponential	$y = ab^x$	Growth/decay, horizontal asymptote
Logarithmic	$y = \log_b x$	Inverse of exponential, vertical asymptote
Trigonometric	$y = \sin x, \cos x, \tan x$	Periodic, amplitude, period

Additional info:

• This review covers nearly all major Precalculus topics, including algebra, functions, graphing, trigonometry, conic sections, sequences, and applications.

• Students should practice solving each type of problem and understand the underlying concepts for exam success.