Chapter 4: Rational, Power, and Root Functions 4.1 Rational Functions and Graphs 4.2 More on Rational Functions and Graphs 4.3 Rational Equations, Inequalities, Models, and Applications 4.4 Functions Defined by Powers and Roots 4.5 Equations, Inequalities, and Applications Involving Root Functions

4.2 More on Rational Functions and Graphs Asymptotes for Rational Functions Let define polynomials. For the rational function defined by written in lowest terms, and for real numbers a and b, 1. If then the line is a vertical asymptote. 2. If then the line is a horizontal asymptote.

4.2 Finding Asymptotes: Example 1 Example 1 Find the asymptotes of the graph of Solution Vertical asymptotes: set denominator equal to 0 and solve.

4.2 Finding Asymptotes: Example 1 Horizontal asymptote: divide each term by the variable factor of greatest degree, in this case x2. Therefore, the line y = 0 is the horizontal asymptote.

4.2 Finding Asymptotes: Example 2 Example 2 Find the asymptotes of the graph of Solution Vertical asymptote: solve the equation x – 3 = 0. Horizontal asymptote: divide each term by x.

4.2 Finding Asymptotes: Example 3 Example 3 Find the asymptotes of the graph of Solution Vertical asymptote: Horizontal asymptote:

4.2 Finding Asymptotes: Example 3 Rewrite f using synthetic division as follows: For very large values of is close to 0, and the graph approaches the line y = x +2. This line is an oblique asymptote (neither vertical nor horizontal) for the graph of the function.

4.2 Determining Asymptotes To find asymptotes of a rational function defined by a rational expression in lowest terms, use the following procedures: Vertical Asymptotes Set the denominator equal to 0 and solve for x. If a is a zero of the denominator but not the numerator, then the line x = a is a vertical asymptote. Other Asymptotes Consider three possibilities: If the numerator has lesser degree than the denominator, there is a horizontal asymptote, y = 0 ( the x-axis). If the numerator and denominator have the same degree and f is of the form

4.2 Determining Asymptotes Other Asymptotes (continued) If the numerator is of degree exactly one greater than the denominator, there may be an oblique (or slant) asymptote. To find it, divide the numerator by the denominator and disregard any remainder. Set the rest of the quotient equal to y to get the equation of the asymptote. Notes: The graph of a rational function may have more than one vertical asymptote, but can not intersect them. The graph of a rational function may have only one other non-vertical asymptote, and may intersect it.

4.2 Comprehensive Graph Criteria for a Rational Function A comprehensive graph of a rational function will exhibits these features: all intercepts, both x and y; location of all asymptotes: vertical, horizontal, and/or oblique; the point at which the graph intersects its non-vertical asymptote (if there is such a point); enough of the graph to exhibit the correct end behavior (i.e. behavior as the graph approaches its nonvertical asymptote).

4.2 Graphing a Rational Function Let define a rational expression in lowest terms. To sketch its graph, follow these steps: Find all asymptotes. Find the x- and y-intercepts. Determine whether the graph will intersect its non-vertical asymptote by solving f (x) = k where y = k is the horizontal asymptote, or f (x) = mx + b where y = mx + b is the equation of the oblique asymptote. Plot a few selected points, as necessary. Choose an x-value between the vertical asymptotes and x-intercepts. Complete the sketch.

4.2 Graphing a Rational Function Example Graph Solution Step 1 Step 2 x-intercept: solve f (x) = 0

4.2 Graphing a Rational Function y-intercept: evaluate f (0) Step 3 To determine if the graph intersects the horizontal asymptote, solve Since the horizontal asymptote is the x-axis, the graph intersects it at the point (–1,0).

4.2 Graphing a Rational Function Step 4 Plot a point in each of the intervals determined by the x-intercepts and vertical asymptotes, to get an idea of how the graph behaves in each region. Step 5 Complete the sketch. The graph approaches its asymptotes as the points become farther away from the origin.

4. 2. Graphing a Rational Function That Does 4.2 Graphing a Rational Function That Does Not Intersect Its Horizontal Asymptote Example Graph Solution Vertical Asymptote: Horizontal Asymptote: x-intercept: y-intercept: Does the graph intersect the horizontal asymptote?

4. 2. Graphing a Rational Function That Does 4.2 Graphing a Rational Function That Does Not Intersect Its Horizontal Asymptote To complete the graph of choose points (–4,1) and .

4.2 Graphing a Rational Function with an Oblique Asymptote Example Graph Solution Vertical asymptote: Oblique asymptote: x-intercept: None since x2 + 1 has no real solutions. y-intercept:

4.2 Graphing a Rational Function with an Oblique Asymptote Does the graph intersect the oblique asymptote? To complete the graph, choose the points

4.2 Graphing a Rational Function with a Hole Example Graph Solution Notice the domain of the function cannot include 2. Rewrite f in lowest terms by factoring the numerator. The graph of f is the graph of the line y = x + 2 with the exception of the point with x-value 2.