Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal

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1 Five-Minute Check (over Lesson 8 3) CCSS Then/Now New Vocabulary Key Concept: Vertical and Horizontal Asymptotes Example 1: Graph with No Horizontal Asymptote Example 2: Real-World Example: Use Graphs of Rational Functions Key Concept: Oblique Asymptotes Example 3: Determine Oblique Asymptotes Key Concept: Point Discontinuity Example 4: Graph with Point Discontinuity 1

2 Over Lesson 8 3 Which is not an asymptote of the function A. x = 6 B. f(x) = 0 C. x = 2 D. f(x) = 4 Which is not an asymptote of the function A. x = 4 B. x = 7 C. x = 4 D. f(x) = 0 State the domain and range of For what value of x is undefined? Over Lesson 8 3 Which is not an asymptote of the function A. x = 6 B. f(x) = 0 C. x = 2 D. f(x) = 4 2

3 Over Lesson 8 3 Which is not an asymptote of the function A. x = 4 B. x = 7 C. x = 4 D. f(x) = 0 Over Lesson 8 3 State the domain and range of A. D = {x x 6} R = {f(x) f(x) 0} B. D = {x x 0} R = {f(x) f(x) 0} C. D = {x x 6, 0} R = {f(x) f(x) 0} D. D = {x x 0} R = {all real numbers} 3

4 Over Lesson 8 3 A. C. B. D. Over Lesson 8 3 For what value of x is undefined? A. 7 and 2 B. 7 and 2 C. 7 and 2 D. 7 and 2 4

5 Content Standards A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Mathematical Practices 7 Look for and make use of structure. You graphed reciprocal functions. Graph rational functions with vertical and horizontal asymptotes. Graph rational functions with oblique asymptotes and point discontinuity. 5

6 rational function vertical asymptote horizontal asymptote oblique asymptote point discontinuity 6

7 Graph with No Horizontal Asymptote Step 1 Find the zeros. x 3 = 0 Set a(x) = 0. x = 0 There is a zero at x = 0. Take the cube root of each side. Graph with No Horizontal Asymptote Step 2 Draw the asymptotes. Find the vertical asymptote. x + 1 = 0 Set b(x) = 0. x = 1 Subtract 1 from each side. There is a vertical asymptote at x = 1. The degree of the numerator is greater than the degree of the denominator. So, there is no horizontal asymptote. 7

8 Graph with No Horizontal Asymptote Step 3 Draw the graph. Use a table to find ordered pairs on the graph. Then connect the points. Graph with No Horizontal Asymptote Answer: 8

9 A. B. C. D. Use Graphs of Rational Functions A. AVERAGE SPEED A boat traveled upstream at r 1 miles per hour. During the return trip to its original starting point, the boat traveled at r 2 miles per hour. The average speed for the entire trip R is given by the formula Draw the graph if r 2 = 15 miles per hour. 9

10 Use Graphs of Rational Functions Original equation r 2 = 15 Simplify. The vertical asymptote is r 1 = 15. Graph the vertical asymptote and function. Notice the horizontal asymptote is R = 30. Answer: Use Graphs of Rational Functions 10

11 Use Graphs of Rational Functions B. What is the R-intercept of the graph? Answer: The R-intercept is 0. Use Graphs of Rational Functions C. What domain and range values are meaningful in the context of the problem? Answer: Values of r 1 greater than or equal to 0 and values of R between 0 and 30 are meaningful. 11

12 A. TRANSPORTATION A train travels at one velocity V 1 for a given amount of time t 1 and then another velocity V 2 for a different amount of time t 2. The average velocity is given by. Let t 1 be the independent variable and let V be the dependent variable. Which graph is represented if V 1 = 60 miles per hour, V 2 = 30 miles per hour, and t 2 = 1 hour? A. B. C. D. 12

13 B. What is the V-intercept of the graph? A. 30 B. 1 C. 1 D. 30 C. What values of t 1 and V are meaningful in the context of the problem? A. t 1 is negative and V is between 30 and 60. B. t 1 is positive and V is between 30 and 60. C. t 1 is positive and V is is less than 60. D. t 1 and V are positive. 13

14 Determine Oblique Asymptotes Graph Step 1 Find the zeros. x 2 = 0 Set a(x) = 0. x = 0 There is a zero at x = 0. Take the square root of each side. 14

15 Determine Oblique Asymptotes Step 2 Find the asymptotes. x + 1 = 0 Set b(x) = 0. x = 1 Subtract 1 from each side. There is a vertical asymptote at x = 1. The degree of the numerator is greater than the degree of the denominator, so there is no horizontal asymptote. The difference between the degree of the numerator and the degree of the denominator is 1, so there is an oblique asymptote. Determine Oblique Asymptotes Divide the numerator by the denominator to determine the equation of the oblique asymptote. 1 ( ) The equation of the asymptote is the quotient excluding any remainder. Thus, the oblique asymptote is the line y = x

16 Determine Oblique Asymptotes Step 3 Draw the asymptotes, and then use a table of values to graph the function. Answer: Graph A. B. C. D. 16

17 Graph. Graph with Point Discontinuity Notice that or x + 2. Therefore, the graph of is the graph of f(x) = x + 2 with a hole at x = 2. 17

18 Graph with Point Discontinuity Answer: Which graph below is the graph of? A. B. C. D. 18

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