Rational Functions and Inequalities

This final page delves into rational functions and inequalities, providing students with tools to analyze and graph these complex mathematical expressions.

Definition: A rational function is a function of the form y = P(x)/Q(x), where P(x) and Q(x) are polynomials and Q(x) ≠ 0.

The guide explains how to graph rational functions, including:

• Identifying vertical and horizontal asymptotes
• Determining the domain and range
• Recognizing reflections and stretches

Example: For y = a/(x-h) + k, the vertical asymptote is at x = h, and the horizontal asymptote is at y = k.

The page also covers more complex rational functions, introducing concepts such as:

• Polynomial asymptotes
• Oblique (slant) asymptotes
• Holes in the graph

Highlight: To find holes in a rational function graph, factor both numerator and denominator. Factors that cancel out indicate potential holes.

The guide concludes with a section on solving systems of rational inequalities:

1. Get zero on one side of the inequality
2. Factor and find the domain
3. Perform sign analysis and graph
4. Declare the solution

This comprehensive coverage of rational functions and inequalities provides students with the tools needed to tackle complex problems in algebra and calculus.

Roots, Radical Notation, and Properties of Exponents

This page covers the basics of roots and radical notation, as well as essential properties of exponents.

Definition: A square root of a number 'a' is a value 'b' such that b² = a. Similarly, an nth root of 'a' is a value 'b' such that bⁿ = a.

Highlight: Odd roots can be applied to any real number, but even roots must be non-negative.

The page also introduces important properties of radicals and exponents, including:

• The product rule for radicals: √(ab) = √a × √b
• The quotient rule for radicals: √(a/b) = √a / √b
• Multiplying conjugates: (a - √b)(a + √b) = a² - b

Example: When simplifying expressions with radicals, using conjugates can be helpful. For instance, to rationalize the denominator of 1/(√3 - 1), multiply both numerator and denominator by (√3 + 1).

The properties of exponents are crucial for working with rational exponents and radicals. Some key properties include:

1. aᵐ × aⁿ = a^(m+n)
2. (aᵐ)ⁿ = a^(mn)
3. (ab)ᵐ = aᵐ × bᵐ

Vocabulary: Rational exponents are exponents that can be expressed as fractions. They provide an alternative way to represent roots.

The page concludes with an explanation of how to convert between radical form and rational exponent form, which is essential for solving radical and rational equations.