Graphing Rational Functions

Oblique Asymptotes

Rational functions may have oblique (slant) asymptotes when the degree of the numerator is exactly one more than the degree of the denominator. To find the equation of the oblique asymptote, perform polynomial long division.

• Definition: An oblique asymptote is a slanted line that the graph of a rational function approaches as x tends to infinity or negative infinity.

• Example: For , divide the numerator by the denominator to find the oblique asymptote.

Introduction to Rational Functions

Domain and Range

The domain of a rational function is all real numbers except where the denominator is zero. The range is all possible output values.

• Example: For , the domain is and the range is also .

Introduction to Logarithms

Graphing Logarithmic Functions

Logarithmic functions have the form or . Their graphs have vertical asymptotes and specific domains and ranges.

• Asymptotes: has a vertical asymptote at .

• Domain: for and .

• Range: All real numbers.

• Example: , .

Graphing Exponential Functions

Exponential Growth

Exponential functions have the form where and . Their graphs pass through (0,1) and have a horizontal asymptote at .

• Example:

• Key Points: (0,1), (1,6), (-1,1/6)

Properties of Logarithms

Logarithmic Equations

Logarithmic properties allow us to simplify and solve equations involving logarithms.

• Key Property:

• Example: Verify if is true. Since , the equation becomes .

Function Composition and Inverses

Inverse Functions

The inverse of a function , denoted , reverses the effect of . To find the inverse, solve for and interchange and .

• Example: For , the inverse is .

Graphing Logarithmic Functions

Solving Logarithmic Equations

To solve equations like , rewrite in exponential form: , then solve for .

The Number e

Exponential Calculations

The number is the base of natural logarithms. Calculators are used to approximate values like to four decimal places.

Quadratic Functions

Graphing and Identifying Quadratics

Quadratic functions have the form . Their graphs are parabolas.

• Vertex Form:

• Example: Given a graph with vertex at (3, -9), find .

Completing the Square

Solving Quadratic Equations

Completing the square is a method to solve quadratic equations by rewriting them in the form .

• Example: Solve by completing the square.

Analytic Geometry

Conic Sections

Equations of the form represent conic sections: parabolas, ellipses, circles, or hyperbolas.

• Example: is a circle if coefficients of and are equal and .

Lines

Point-Slope and Slope-Intercept Form

The equation of a line can be written in point-slope form and converted to slope-intercept form .

• Example: Find the equation of a line passing through and .

Linear Equations

Solving Linear Equations

Linear equations can be solved by isolating the variable using algebraic operations.

• Example: Solve for .

Completing the Square and Quadratic Formula

Quadratic Formula

The quadratic formula solves any quadratic equation .

• Example: Solve using the quadratic formula.

Complex Numbers

Operations with Complex Numbers

Complex numbers are of the form , where . Operations include addition, subtraction, and multiplication.

• Example:

Quadratic Expressions and Undefined Values

Undefined Expressions

Expressions involving square roots or denominators are undefined for values that make the radicand negative or the denominator zero.

• Example: is undefined when .

Linear Inequalities and Domains

Domain of Square Root Functions

The domain of is the set of values for which .

• Example: is defined when .

Multiplying and Factoring Polynomials

Polynomial Operations

Multiplying and factoring polynomials are essential algebraic skills in precalculus.

• Example: Expand or factor .

Rational Exponents and Simplifying Expressions

Simplifying Rational Expressions

To simplify rational expressions, factor numerator and denominator and cancel common factors.

• Example: Simplify .