What is the end behavior? Rational functions may have asymptotes. The function f(x) = has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0. 1 x Parent Function What is the end behavior? What are the intervals of increasing and decreasing? Slide 1 of 25

The rational function f(x) = can be transformed by using methods similar to those used to transform other types of functions. 1 x Slide 2 of 25

Example 1: Transforming Rational Functions Using the graph of f(x) = as a guide, describe the transformation and graph each function. 1 x + 2 1 x – 3 A. g(x) = B. g(x) = Because h = –2, translate f 2 units left. Because k = –3, translate f 3 units down. New Asymptotes, EB and inc/dec? Slide 3 of 25

Because k = 1, translate f 1 unit up. Check It Out! Example 1 Using the graph of f(x) = as a guide, describe the transformation and graph each function. 1 x 1 x + 4 1 x + 1 a. g(x) = b. g(x) = Because k = 1, translate f 1 unit up. Because h = –4, translate f 4 units left. New Asymptotes, EB and inc/dec? Slide 4 of 25

The values of h and k affect the locations of the asymptotes, the domain, and the range of rational functions whose graphs are hyperbolas. Slide 5 of 25

Example 2: Determining Properties of Hyperbolas Identify the asymptotes, domain, and range of the function g(x) = – 2. 1 x + 3 1 x – (–3) g(x) = – 2 h = –3, k = –2. Vertical asymptote: x = –3 The value of h is –3. Domain: {x|x ≠ –3} Horizontal asymptote: y = –2 The value of k is –2. Range: {y|y ≠ –2} Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = –3 and y = –2. Slide 6 of 25

Identify the asymptotes, domain, and range of the function g(x) = – 5. Check It Out! Example 2 Identify the asymptotes, domain, and range of the function g(x) = – 5. 1 x – 3 1 x – (3) g(x) = – 5 h = 3, k = –5. Vertical asymptote: x = 3 The value of h is 3. Domain: {x|x ≠ 3} Horizontal asymptote: y = –5 The value of k is –5. Range: {y|y ≠ –5} Check Graph the function on a graphing calculator. The graph suggests that the function has asymptotes at x = 3 and y = –5. Slide 7 of 25

A continuous function is a function whose graph has no gaps or breaks. A discontinuous function is a function whose graph has one or more gaps or breaks. A continuous function is a function whose graph has no gaps or breaks. Slide 8 of 25

Set the top=0 Set the bottom=0 Slide 10 of 25

Some rational functions have a horizontal asymptote Some rational functions have a horizontal asymptote. The existence and location of a horizontal asymptote depends on the degrees of the polynomials that make up the rational function. Note that the graph of a rational function can sometimes cross a horizontal asymptote. Slide 11 of 25

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Slant Asymptote If the degree of the top is one bigger than the degree of the bottom then there is a slant asymptote. You will find it by doing long division until you get y = mx + b

Identify the zeros and asymptotes of the function. x – 2 x2 – 1 f(x) = x – 2 (x – 1)(x + 1) f(x) = Zero: 2 Vertical asymptote: x = 1, x = –1 Horizontal asymptote: y = 0 Slide 13 of 25

Identify the zeros and asymptotes of the function. 4x – 12 x – 1 f(x) = 4(x – 3) x – 1 f(x) = Zero: 3 Vertical asymptote: x = 1 Horizontal asymptote: y = 4 Slide 14 of 25

Identify the zeros and asymptotes, y-int of the function. Then graph. x2 + 2x – 15 x – 1 f(x) = (x – 3)(x + 5) x – 1 f(x) = Zeros: 3 and –5 Vertical asymptote: x = 1 Horizontal asymptote: none Slant asymptote: y= x + 3 Y-int: (0, 15) EB and inc/dec? Slide 15 of 25

Check It Out! Example 4a Continued Identify the zeros and asymptotes of the function. Then graph. Graph by using a table of values. Slide 16 of 25

Identify the zeros, y-int and asymptotes of the function. Then graph. 3x2 + x x2 – 9 f(x) = x(3x – 1) (x – 3) (x + 3) f(x) = Zeros: 0 and – 1 3 Vertical asymptote: x = –3, x = 3 Horizontal asymptote: y = 3 Y-int: (0, 0) EB and inc/dec? Slide 17 of 25

Check It Out! Example 4c Continued Graph by using a table of values. Slide 18 of 25

VA always wins Slide 19 of 25

Example 5: Graphing Rational Functions with Holes Identify holes in the graph of f(x) = . x2 – 9 x – 3 (x – 3)(x + 3) x – 3 f(x) = There is a hole in the graph at x = 3. To find where the hole would be, plug in 3 into the reduced equation. f(3) = 3+3 therefore hole is at (3, 6) Slide 20 of 25

Find all zeros, y-int, holes, asymptotes and domain Slide 21 of 25

Identify the domain, zeros, y-int, asymptotes, and holes in the graph of a rational function. Then graph f(x) = 2x2 + 2x x2 – 1 factor: f(x) = 2x(x + 1) (x + 1)(x – 1) Slide 5 of 11 Slide 22 of 25

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zero: 2; asymptotes: x = 0, y = 1; hole at x = 1 Lesson Quiz: Part II 3. Identify the zeros, asymptotes, and holes in the graph of . Then graph. x2 – 3x + 2 x2 – x f(x) = zero: 2; asymptotes: x = 0, y = 1; hole at x = 1 Slide 24 of 25

Steps include: Find zeros, VA, HA, SA, holes, HW: P 597 19-22 Describe transformations, any asymptotes, D & R, EB, inc/dec and then graph, then do all steps on 23, 25, 27, 31, 33, then on 34-37 (don’t graph any) Steps include: Find zeros, VA, HA, SA, holes, y –int, Domain, EB and inc/dec. Plot on graph then use the table from the calculator Slide 25 of 25