Pre-AP Algebra 2 Unit 8 - Lesson 2 Graphing rational functions by plugging in numbers; feature analysis

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1 Pre-AP Algebra 2 Unit 8 - Lesson 2 Graphing rational functions by plugging in numbers; feature analysis Objectives: Students will be able to: Analyze the features of a rational function: determine domain, vertical asymptotes, x- and y-intercepts Graph simple rational functions (with no holes) by plugging in values for x and plotting the points Make a quick sketch of a function, given a set of features Materials: Hw #8-1 answers overhead; tally sheets; pair work; note-taking templates; homework #8-2 Time Activity 5 min Check Homework Students check their answers to hw #8-1 on the overhead. Pass around the tally sheets. 10 min Review Homework Review the top 2 3 problems from the homework. Problems to grade: 1d, 2, 3a, 3c, 3f 40 min Pair Work Given a pair of lines, students are asked to analyze the quotient (with each line as both the numerator and denominator) by finding the intercepts and the vertical asymptote. Then, they plug in points and graph the result. Finally, they check their graphs by graphing on the calculator. 25 min Direct Instruction Background: The domain of a function is the set of all values of x that the function exists at (is defined at). Concepts: A rational function as a function written in the form gx ( ), where g and h are polynomials, hx ( ) and h(x) 0. x-intercepts occur when y = 0. That means f(x) = 0. This only happens in a fraction when the numerator is zero. To find x-intercepts: set g(x) = 0 and solve. The y-intercept occurs when x = 0. To find y-intercept: calculate f(0). Why can there only be one y-intercept on any function, but many x-intercepts? Vertical asymptotes occur when h(x) = 0. To find v.a.: set h(x) = 0 and solve The domain of a rational function is all real numbers except the solutions to h(x) = 0. This is true because, when you divide by 0, the result is undefined. Examples: For each function, find the intercepts, asymptotes, and domain. Make a quick sketch based on this information. Then, graph on the calculator to check. a) 3x 4 2x 5 b) f (x) 8 x 2 6x 16 c) f (x) x 6 x d) f (x) x 6 x 2 4 Homework #8-2: Graphing Rational Functions

2 Pre-AP Algebra 2 Pair Work Name: Graphing Rational Functions For the following problems, let f(x) = 2x 3 and let g(x) = x ) Let r(x) f (x) (fill in the blanks with the correct expressions). g(x) a. When does a vertical asymptote occur in any graph? Why does it happen? b. What is the equation of the vertical asymptote of r(x)? c. Where do x-intercepts occur on any graph? If the function is a fraction, what must be true for this to happen? d. What is the x-intercept of r(x)? e. Where does the y-intercept occur on any graph? f. What is the y-intercept of r(x)? g. Plot the x- and y-intercepts, and make a dotted line for the vertical asymptote. Then, fill in the table and plot the rest of the points to complete the hyperbola. x r(x) x r(x)

3 2) Let q(x) g(x) (fill in the blanks with the correct expressions). f (x) a. What is the equation of the vertical asymptote? b. What is the x-intercept? c. What is the y-intercept? d. Plot the intercepts and make a dotted line for the vertical asymptotes. Fill in the table and graph the hyperbola. x q(x) x q(x) ) Set your calculator s viewing window to 10 10, like the above axes. Set Y1 = 2x 3 and then set Y2 = x + 2. Deselect these equations (move to the equal sign and press ENTER to turn off the highlight). Move the cursor to Y3 and press VARS. Press RIGHT ARROW and then choose FUNCTION. Press enter on Y1. Press the divide key. Now press VARS, RIGHT ARROW, FUNCTION again and choose Y2. You should now see Y3 = Y1/Y2. Press GRAPH and you should have a picture that looks like your first graph. (Note: you may see a vertical line this is an error the calculator makes due to the 0 in the denominator. It s not really part of the graph.) 4) Repeat this process to graph the second function. Your Y3 should equal Y2/Y1. 5) Look back at your graphs. a. What is the domain of r(x)? b. What is the domain of q(x)? 6) On your calculator, if you change Y3 to read Y3 = Y1 * Y2, what kind of graph do you think you ll get? Why? Try it and see if you are correct.

4 Pre-AP Algebra 2 Homework #8-2 Name: Hw #8-2: Graphing Rational Functions 1) Determine the features of the graph of f(x) shown to the right. a. Vertical asymptotes: b. x-intercepts: c. y-intercept: d. Domain: 2) Make a sketch of a rational function that has vertical asymptotes at x = -2 and x = 3, a y-intercept at (0, 4), and x-intercepts at (-5, 0) and (6, 0). Remember that a vertical asymptote can never be crossed by the function s graph. Write the domain of your graph: 3) The hyperbola shown here was generated by dividing the two lines. Determine which line is the numerator and which is the denominator. Explain clearly how you are able to tell.

5 4) Given the function x 1 : x 3 a. Determine the x- and y-intercepts, vertical asymptote, and the domain. b. What is the name of the kind of graph that will be generated? c. Plot the intercepts on the graph and draw in a dotted line for the vertical asymptote. Complete the table and then make the graph. x f(x) 5) For each function, find the x- and y-intercepts, vertical asymptotes, and the domain. Then, graph the function on the calculator to verify your answers. a. 2 x 1 x-intercepts: y-intercept: b. f (x) x 2 x 2 25 x 2 x 20 c. f (x) 2x 3 3x 2 18x 27 d. f (x) x3 8 x 2 16 x-intercepts: x-intercepts: x-intercepts: y-intercept: y-intercept: y-intercept:

6 Hw #8-1 Tally Sheet 1) a) x = -2 b) As x -2 -, f(x) As x -2 +, f(x) - c)(3, 0) d) n(x) is line B, d(x) is line A 2) Answers will vary 3) a) V.A.: x = -3 b) V.A.: x = 9 c) V.A.: x = 1/4 x-int: (5/2, 0) x-int: (-2, 0) x-int: none d) V.A.: none e) V.A.: x = 4, x = -3 x-int: (4, 0), (-4, 0) x-int: (-5, 0) f) V.A.: x = 2, x = -2 x-int: (3, 0)

7 Hw #8-1 Tally Sheet 1) a) b) c) d) 2) 3) a) b) c) d) e) f)