Precalculus Exam Review: Factoring, Quadratics, Circles, Rational and Exponential Functions

R.4 – Factoring

Factoring Polynomial Expressions

Factoring is the process of expressing a polynomial as a product of its factors. This is a foundational skill in algebra and precalculus, useful for solving equations and simplifying expressions.

• Factoring Quadratics: Express $ax^2 + bx + c$ as $(mx + n)(px + q)$ where $m, n, p, q$ are constants.

• Factoring by Grouping: Used for polynomials with four or more terms.

• Special Products: Recognize patterns such as difference of squares: $a^2 - b^2 = (a - b)(a + b)$.

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

1.4 – Quadratic Equations

Solving Quadratic Equations

Quadratic equations are equations of the form $ax^2 + bx + c = 0$. There are three main methods to solve them: factoring, completing the square, and using the quadratic formula.

• Factoring: Set the equation to zero and factor.

• Completing the Square: Rearrange and add terms to form a perfect square trinomial.

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

• Example: Solve $x^2 + 8x = 20$ by all three methods.

2.2 – Circles

Equations and Properties of Circles

The standard equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$.

• Finding Center and Radius: Rearrange the equation to standard form.

• General Form: $x^2 + y^2 + Dx + Ey + F = 0$

• Finding Equation Given Center and Point: Use the distance formula to find $r$.

• Example: Find the center and radius of $(x - 3)^2 + y^2 = 64$.

3.1 – Quadratic Functions

Analyzing Quadratic Functions

Quadratic functions are of the form $f(x) = ax^2 + bx + c$. Key features include domain, range, vertex, axis of symmetry, and intercepts.

• Domain: All real numbers, $(-\infty, \infty)$.

• Range: Depends on the vertex and direction of opening.

• Vertex: $x = -\frac{b}{2a}$, $y = f(-\frac{b}{2a})$

• Axis of Symmetry: $x = -\frac{b}{2a}$

• Intercepts: Set $x = 0$ for $y$-intercept, $f(x) = 0$ for $x$-intercepts.

• Example: For $f(x) = x^2 - 6x - 12$, find all features.

R.5 – Rational Expressions

Operations with Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. Key operations include addition, subtraction, multiplication, division, and finding difference quotients.

• Difference Quotient: $\frac{f(x+h) - f(x)}{h}$

• Simplifying: Factor and reduce common terms.

• Adding/Subtracting: Find common denominators.

• Example: Add $\frac{2}{x+4} + \frac{4}{x+2}$ and simplify.

1.6 – Rational Equations

Solving Rational Equations

Rational equations contain rational expressions set equal to each other. Solutions require finding a common denominator and checking for extraneous solutions.

• Clear Denominators: Multiply both sides by the least common denominator.

• Check Solutions: Substitute back to avoid division by zero.

• Example: Solve $\frac{3}{x-2} = \frac{2}{x+4}$.

4.1 – Inverse Functions

Finding and Analyzing Inverse Functions

An inverse function reverses the effect of the original function. Not all functions have inverses; a function must be one-to-one.

• Finding the Inverse: Swap $x$ and $y$ and solve for $y$.

• Domain and Range: The domain of $f$ becomes the range of $f^{-1}$ and vice versa.

• One-to-One Test: A function passes the horizontal line test if it has an inverse.

• Example: Find the inverse of $f(x) = (x - 2)^3 + 7$.

4.2 – Exponents and Exponential Functions

Properties and Graphs of Exponential Functions

Exponential functions have the form $f(x) = a^x$. They model growth and decay and have unique properties regarding domain, range, and asymptotes.

• Domain: All real numbers, $(-\infty, \infty)$.

• Range: $(0, \infty)$ for $a > 0$.

• Horizontal Asymptote: Typically $y = 0$ or shifted by constants.

• Intercepts: $x$-intercept: solve $a^x = 0$ (may not exist); $y$-intercept: $f(0)$.

• Example: For $f(x) = 5^{x+2} - 1$, find intercepts.

Summary Table: Key Features of Functions

Function Type	Domain	Range	Intercepts	Asymptotes
Quadratic	$(-\infty, \infty)$	Depends on vertex	Find by setting $x=0$ and $f(x)=0$	None
Rational	Exclude zeros of denominator	Depends on numerator/denominator	Set numerator/denominator to zero	Vertical: zeros of denominator; Horizontal: degree comparison
Exponential	$(-\infty, \infty)$	$(0, \infty)$	$y$-intercept: $f(0)$	Horizontal: $y =$ constant

Additional info:

• This review covers essential Precalculus topics for exam preparation, including factoring, solving quadratics, properties of circles, rational and exponential functions, and inverse functions.

• Each topic includes definitions, formulas, and examples to reinforce understanding.