The Parent Function can be transformed by using What do a, h and k represent? a the vertical stretch or compression h the horizontal translation k the vertical translation

There can be vertical asymptotes horizontal asymptotes What is an asymptote? The line where a graph approaches There can be vertical asymptotes horizontal asymptotes and slant asymptotes. What is the vertical asymptote? What is the horizontal asymptote?

Example 1 What is the transformation? What is the vertical asymptote? Down 3 units What is the vertical asymptote? What is the horizontal asymptote?

Example 1 What is the Domain? What is the Range?

Example 2 What is the transformation? What is the vertical asymptote? Left 2 units What is the vertical asymptote? What is the horizontal asymptote?

To Graph Rational Functions Factor Completely Vertical Asymptotes: set the denominator equal to zero. It is stated as an equation The vertical asymptotes are the domain exclusions. 3. Horizontal Asymptotes: compare the degrees of the numerator and denominator.

Horizontal Asymptote If then HA: y = 0 If then HA: y = 2 If HA: none you look at then HA: y = 2 If then there is NO horizontal asymptote HA: none

Graphing continued What is another name for the x-intercepts? 4. Roots: set the numerator = 0 What is another name for the x-intercepts? Roots What does it mean on the graph? Where the graph crosses the x-axis

Graphing continued 5. Holes: Look at factors that are common to numerator and denominator. Set those equal to 0 and solve. Vertical Asymptotes do NOT include the holes.

8-4: Graphs of Rational Functions [Day 1] Example 3 Example 2 Identify the following for Vertical asymptote Horizontal Asymptote Roots: Holes: Domain: Range: Y-intercept: 11/29/2018 1:50 AM 8-4: Graphs of Rational Functions [Day 1] 10 10

8-4: Graphs of Rational Functions [Day 1] Example 4 Identify the following for Vertical asymptote Horizontal Asymptote Roots: Holes: Y-intercept: Vertical asymptote Roots Horizontal Asymptote look at the degrees Same Degree Divide the leading coefficients. 11/29/2018 1:50 AM 8-4: Graphs of Rational Functions [Day 1] 11 11

8-4: Graphs of Rational Functions [Day 1] Example 4 graph Zeros: Vertical Asymptote: Horizontal Asymptote: Domain: Range: x = –2 y = 1 8-4: Graphs of Rational Functions [Day 1] 12

8-4: Graphs of Rational Functions [Day 1] Example 5 Identify the following for Vertical asymptote Roots Vertical asymptote Horizontal Asymptote Roots: Holes: Y-intercept: Horizontal Asymptote Holes look at the degrees 8-4: Graphs of Rational Functions [Day 1] 11/29/2018 1:50 AM 13 13

8.4 - Graphs of Rational Functions [Day 2] Example 5 11/29/2018 1:50 AM 8.4 - Graphs of Rational Functions [Day 2] 14 14

8-4: Graphs of Rational Functions [Day 1] Example 6 Identify the following for Vertical asymptote Roots Vertical asymptote Horizontal Asymptote Roots: Holes: Horizontal Asymptote Holes look at the degrees 8-4: Graphs of Rational Functions [Day 1] 11/29/2018 1:50 AM 15 15

8.4 - Graphs of Rational Functions [Day 2] Example 6 11/29/2018 1:50 AM 8.4 - Graphs of Rational Functions [Day 2] 16 16

Extra problems following this slide

8.4 - Graphs of Rational Functions [Day 2] Example 7 Given, graph, determine the zeros, all asymptotes/holes, and domain 11/29/2018 1:50 AM 8.4 - Graphs of Rational Functions [Day 2] 18 18

8.4 - Graphs of Rational Functions [Day 2] Example 8 Given, graph, determine the zeros and all asymptotes/holes Hole at x = 3 11/29/2018 1:50 AM 8.4 - Graphs of Rational Functions [Day 2] 19 19

8.4 - Graphs of Rational Functions [Day 2] Without a Calculator Identify the zeros and determine all of the asymptotes to the following: 4x – 12 x – 1 1. f(x) = (x + 2)(x – 3) x + 1 2. f(x) = x – 2 x2 + x 3. f(x) = 11/29/2018 1:50 AM 8.4 - Graphs of Rational Functions [Day 2] 20 20

8-4: Graphs of Rational Functions [Day 1] Example 3 Your Turn Your Turn Identify the zeros and vertical asymptote(s) of 11/29/2018 1:50 AM 8-4: Graphs of Rational Functions [Day 1] 21 21

Horizontal Asymptotes: compare the degrees of the numerator and denominator. Degree of Numerator > Degree of Denominator Top bigger---NO…means NO horizontal asymptotes. Must find Oblique asymptote. Degree of Denominator > Degree of Numerator Bottom bigger---OH…means horizontal asymptote is y=0. Degree of Numerator = Degree of Denominator -CO…means find ratio of leading coefficients

8-4: Graphs of Rational Functions [Day 1] Example 5 Identify the following for Vertical asymptote Roots Vertical asymptote Horizontal Asymptote Roots: Holes: Horizontal Asymptote look at the degrees Same Degree Divide the leading coefficients. 8-4: Graphs of Rational Functions [Day 1] 11/29/2018 1:50 AM 23 23

8-4: Graphs of Rational Functions [Day 1] Example 5 Given Zeros: Vertical Asymptote: Horizontal Asymptote: Domain: Range: x = –2 x = 2 y = 2 11/29/2018 1:50 AM 8-4: Graphs of Rational Functions [Day 1] 24 24