Warm-Up: FACTOR x2 – 36 5x2 - 20 3x + 7 x2 – x – 2 x2 – 5x – 14

Answers (x + 6)(x – 6) 5(x + 2)(x – 2) Unfactorable (x – 2) (x + 1)

GRAPHING RATIONAL FUNCTIONS A rational function has a rule given by a fraction whose numerator and denominator are polynomials, and whose denominator is not 0. EXAMPLES:

Characteristics of Rational Functions Domain Range X-intercepts (zeros) Y-intercept Interval of Increase Interval of Decrease Local Extrema (Minima, Maxima) Vertical Asymptote Horizontal Asymptote Slant Asymptote Holes

An asymptote is a line that a graph approaches more and more closely. EXAMPLES: Vertical asymptote

Domain Domain Set of “input” values The first coordinate of an ordered pair (DOMAIN, range) To find the limitations on the domain of rational functions, set the denominator ≠ 0. These are the value(s) that are NOT in the domain.

Find the domain

Find the domain

Find the domain

Finding x-intercepts Set the numerator= 0. Factor. Solve for x.

Find the x-intercept

Find the domain

Find the domain

Finding y-intercepts Plug in ‘0’ for all x-values. Solve for y.

Find the y-intercept

Find the y-intercept

Find the y-intercept

Finding Vertical Asymptotes and Holes in the Graph 1. Factor if possible and cancel identical terms Set the cancelled term = 0 …this is a hole in the graph 2. Set the remaining term in the denominator = 0 and solve to find the V.A.

Example Hole: x = -1; (-1, 1) Vertical Asym: x = -2

Example Hole: NONE Vertical Asym: x = -5/3

Example Hole: x = -2; (-2, 1/2) Vertical Asym: x = -4

Write the equations for the vertical asymptotes.

Find the holes and equations for the vertical asymptotes. No holes. No holes. No holes. No holes.

Identifying Degrees of Functions What is the degree of each function? 2x2 – x – 1 2x – 1 x3 – 5x + 6 x3 – 2x2 + 7x – 1 2 1 3 3

Identifying Horizontal Asymptotes Compare the degrees of the polynomials in the numerator and denominator to find the horizontal asymptotes of a function.

If the degree of the numerator and denominator are equal, then is the H.A. If the degree of the NUMERATOR is GREATER than the degree of denominator, then there is NO H.A. If the degree of the NUMERATOR is LESS than the degree of the denominator, then is the H.A.

Example

Example

Example No Horizontal Asymptote

Write the equations for the horizontal asymptotes.

Write the equations for the horizontal asymptotes. No H.A. No H.A.

Slant Asymptotes

Slant Asymptotes If the degree of the numerator is EXACTLY ONE MORE than the degree of the denominator, then the graph has a slant asymptote.

How to Find the Equation of a Slant Asymptote Use long division Ignore the remainder The equation that is your answer after dividing is the equation of the slant asymptote.

Write the equation of the slant asymptote. y = x No Slant. Degrees are equal. No Slant. Degree of numerator is smaller. y = x - 2

Graphing Rational Functions Find the VA. Sketch a vertical dotted line for the VA. Find the HA. Sketch a horizontal dotted line for the HA. Find the SLA (if any). Sketch a dotted line for the SLA. Find the x-int by setting the numerator = 0. Find the y-int by setting x = 0. Find additional points on both sides of the x value(s) of the VA.

In the following problems, a) identify all intercepts, b) find any vertical and horizontal asympotes, c) plot additional solution points, d) sketch the graph.

A) Identify all intercepts B) Find all vertical and slant asymptotes C) Plot additional points as needed D) Sketch the graph

A) Identify all intercepts B) Find all vertical and slant asymptotes C) Plot additional points as needed D) Sketch the graph

A) Identify all intercepts B) Find all vertical and slant asymptotes C) Plot additional points as needed D) Sketch the graph