You graphed reciprocal functions Graph rational functions with

You graphed reciprocal functions. • Graph rational functions with vertical and horizontal asymptotes. • Graph rational functions with oblique asymptotes and point discontinuity.

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

Graph with No Horizontal Asymptote Step 1 Find the zeros. x 3 = 0 x = 0 Set a(x) = 0. Take the cube root of each side. There is a zero at x = 0.

Graph with No Horizontal Asymptote Step 2 Draw the asymptotes. Find the vertical asymptote. x+1 = 0 x = – 1 Set b(x) = 0. 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.

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:

Graph with No Horizontal Asymptote Answer:

A. B. C. D.

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 per hour. Draw the graph if r 2 = 15 miles

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.

Use Graphs of Rational Functions Answer:

Use Graphs of Rational Functions Answer:

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

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:

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.

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.

A. B. C. D.

B. What is the V-intercept of the graph? A. 30 B. 1 C. – 1 D. – 30

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.

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.

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

Determine Oblique Asymptotes Step 2 Find the asymptotes. x+1 = 0 x = – 1 Set b(x) = 0. 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 (–) 1 The equation of the asymptote is the quotient excluding any remainder. Thus, the oblique asymptote is the line y = x – 1.

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

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

Graph A. B. C. D.

Graph A. B. C. D.

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

Graph with Point Discontinuity Answer:

Graph with Point Discontinuity Answer:

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

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