Precalculus Exam Solutions and Study Guide

Fundamental Concepts of Algebra

Simplifying Expressions

Algebraic simplification involves reducing expressions to their simplest form using arithmetic operations and properties of exponents and radicals.

• Example: Simplify $\sqrt{25/3}$.

• Key Point: Rationalizing denominators is often required for final answers.

• Formula: $\sqrt{a/b} = \frac{\sqrt{a}}{\sqrt{b}}$

Factoring

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

• Key Point: Common techniques include factoring trinomials, difference of squares, and sum/difference of cubes.

• Example: Factor $x^2 - 9x + 6$ as $(x-9)(x+6)$.

• Formula: $a^2 - b^2 = (a-b)(a+b)$

Functions and Graphs

Domain and Range

The domain of a function is the set of all possible input values, while the range is the set of all possible output values.

• Example: For $f(x) = \sqrt{x-7}$, $D: x \geq 7$, $R: y \geq 0$.

Transformations of Functions

Transformations include translations, reflections, stretches, and compressions of graphs.

• Key Point: $y = f(x-h) + k$ shifts the graph $h$ units right and $k$ units up.

• Example: Graph $y = -2|x-1|$ by reflecting, stretching, and translating the parent function $y = |x|$.

Polynomial and Rational Functions

Multiplying and Simplifying Polynomials

Use distributive property and combine like terms to multiply polynomials.

• Example: $(x+2)(x+5) = x^2 + 7x + 10$

Dividing Rational Expressions

To divide rational expressions, multiply by the reciprocal of the divisor.

• Formula: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$

• Example: $\frac{2}{x(x+2)}$

Systems of Equations and Matrices

Solving Systems of Equations

Systems can be solved by substitution, elimination, or using matrices.

• Example: Solve $2x + 3y = 2$, $x - 3y = 4$.

Augmented Matrices

An augmented matrix represents a system of linear equations for solution via row operations.

Equation	Matrix Row
$2x + 3y = 2$	2 3 | 2
$x - 3y = 4$	1 -3 | 4

Conic Sections and Analytic Geometry

Circles

The equation of a circle in standard form is $(x-h)^2 + (y-k)^2 = r^2$.

• Center: $(h, k)$

• Radius: $r$

• Example: $(x-2)^2 + (y+1)^2 = 9$ has center $(2, -1)$ and radius $3$.

Midpoint Formula

The midpoint $M$ of a segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is:

• Formula: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Sequences, Induction, and Probability

Arithmetic Sequences

An arithmetic sequence has a constant difference between consecutive terms.

• Formula: $a_n = a_1 + (n-1)d$

Additional Topics

Word Problems and Applications

Precalculus often includes real-world applications such as cost analysis, projectile motion, and geometric problems.

• Example: Comparing internet service plans using systems of equations.

• Example: Modeling the height of a dropped object: $s(t) = s_0 - 32t$

Graphing and Slope

The slope of a line perpendicular to $y = mx + b$ is $-1/m$.

• Formula: $m_{\text{perp}} = -\frac{1}{m}$

Summary Table: Key Algebraic Formulas

Concept	Formula
Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Distance Formula	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
Midpoint Formula	$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$
Circle Equation	$(x-h)^2 + (y-k)^2 = r^2$