Graphing Rational Functions LESSON 8–4 Graphing Rational Functions

Five-Minute Check (over Lesson 8–3) TEKS 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 Lesson Menu

Which is not an asymptote of the function A. x = –6 B. f(x) = 0 C. x = 2 D. f(x) = 4 5-Minute Check 1

Which is not an asymptote of the function A. x = –4 B. x = 7 C. x = 4 D. f(x) = 0 5-Minute Check 2

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} 5-Minute Check 3

A. B. C. D. 5-Minute Check 4

For what value of x is undefined? A. –7 and –2 B. 7 and –2 C. –7 and 2 D. 7 and 2 5-Minute Check 5

Mathematical Processes A2.1(E), A2.1(F) Targeted TEKS A2.6(K) Determine the asymptotic restrictions on the domain of a rational function and represent domain and range using interval notation, inequalities, and set notation. Mathematical Processes A2.1(E), A2.1(F) TEKS

You graphed reciprocal functions. Graph rational functions with vertical and horizontal asymptotes. Graph rational functions with oblique asymptotes and point discontinuity. Then/Now

rational function vertical asymptote horizontal asymptote oblique asymptote point discontinuity Vocabulary

Concept

x = 0 Take the cube root of each side. There is a zero at x = 0. Graph with No Horizontal Asymptote Step 1 Find the zeros. x3 = 0 Set a(x) = 0. x = 0 Take the cube root of each side. There is a zero at x = 0. Example 1

Step 2 Draw the asymptotes. Find the vertical asymptote. 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. Example 1

Graph with No Horizontal Asymptote Step 3 Draw the graph. Use a table to find ordered pairs on the graph. Then connect the points. Example 1

Graph with No Horizontal Asymptote Answer: Example 1

B. C. D. Example 1

Use Graphs of Rational Functions A. AVERAGE SPEED A boat traveled upstream at r1 miles per hour. During the return trip to its original starting point, the boat traveled at r2 miles per hour. The average speed for the entire trip R is given by the formula Draw the graph if r2 = 15 miles per hour. Example 2A

Original equation r2 = 15 Simplify. Use Graphs of Rational Functions Original equation r2 = 15 Simplify. The vertical asymptote is r1 = –15. Graph the vertical asymptote and function. Notice the horizontal asymptote is R = 30. Example 2A

Use Graphs of Rational Functions Answer: Example 2A

B. What is the R-intercept of the graph? Use Graphs of Rational Functions B. What is the R-intercept of the graph? Answer: The R-intercept is 0. Example 2B

Use Graphs of Rational Functions C. What domain and range values are meaningful in the context of the problem? Answer: Values of r1 greater than or equal to 0 and values of R between 0 and 30 are meaningful. Example 2C

A. TRANSPORTATION A train travels at one velocity V1 for a given amount of time t1 and then another velocity V2 for a different amount of time t2. The average velocity is given by . Let t1 be the independent variable and let V be the dependent variable. Which graph is represented if V1 = 60 miles per hour, V2 = 30 miles per hour, and t2 = 1 hour? Example 2

A. B. C. D. Example 2

B. What is the V-intercept of the graph? D. –30 Example 2

A. t1 is negative and V is between 30 and 60. C. What values of t1 and V are meaningful in the context of the problem? A. t1 is negative and V is between 30 and 60. B. t1 is positive and V is between 30 and 60. C. t1 is positive and V is is less than 60. D. t1 and V are positive. Example 2

Concept

x = 0 Take the square root of each side. There is a zero at x = 0. Determine Oblique Asymptotes Graph Step 1 Find the zeros. x2 = 0 Set a(x) = 0. x = 0 Take the square root of each side. There is a zero at x = 0. Example 3

Step 2 Find the asymptotes. x + 1 = 0 Set b(x) = 0. 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. Example 3

The equation of the asymptote is the quotient excluding any remainder. Determine Oblique Asymptotes Divide the numerator by the denominator to determine the equation of the oblique asymptote. – 1 (–) 1 The equation of the asymptote is the quotient excluding any remainder. Thus, the oblique asymptote is the line y = x – 1. Example 3

Determine Oblique Asymptotes Step 3 Draw the asymptotes, and then use a table of values to graph the function. Answer: Example 3

Graph B. C. D. Example 3

Concept

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. Example 4

Graph with Point Discontinuity Answer: Example 4

Which graph below is the graph of ? A. B. C. D. Example 4

Graphing Rational Functions LESSON 8–4 Graphing Rational Functions