Exploring Rational Functions

Transcription

1 Name Date Period Exploring Rational Functions Part I - The numerator is a constant and the denominator is a linear factor. 1. The parent function for rational functions is: Graph and analyze this function: Domain: Range: 2. Graph the following transformed functions: Generalize the patterns you see by answering: How is transformed from the parent function? Discuss the transformations and any changes to the analysis of the graph. In particular: Where do the vertical asymptotes occur? 3. Graph the following transformed functions: = Generalize the patterns you see by answering: How is transformed from the parent function? Discuss the transformations and any changes to the analysis of the graph. 4. Graph the following transformations: Generalize the patterns you see by answering: How is + k transformed from the parent function? Discuss the transformations and any changes to the analysis of the graph (in particular, asymptotes). Part II Higher order polynomials in the denominator 1. Why are the two functions below the same? Use algebra (factoring) to show they are the same function.

2 Practice: Write each function with a factored denominator. 2. Now graph all five functions: How do the vertical asymptotes (plural!) relate to the functions factored denominators? Generalize your results by completing the sentence: Given a rational function in the form, the function s vertical asymptotes will be. Part III - Zeros of rational functions 1. Recall how fractions work : Evaluate the following without a calculator: Why did you get the same result when you evaluated each fraction? 2. For each function below, find the x-intercepts (zeros) of the function. (Hint: we only need the numerator to be 0.) You may use algebra or a graphing calculator. Then generalize the patterns you see by completing the sentence. Given a rational function in the form: r(x) =, then the s of are the choose f(x) or g(x) zeros of the rational function, r(x), as well. Why does this result make sense?

3 Part IV - Exploring horizontal and slant asymptotes (which describe the END BEHAVIOR of the functions) The degree of the numerator is often referred to as m and the degree of the denominator as n. Examples: m = 2 and n = 1 m = 3 and n = 2 1. For each function below, state m and n. Then graph and state the horizontal asymptote. Function m n equation of the horizontal asymptote In each case above, m n and the horizontal asymptote is. choose <, >, or = Generalize by completing the sentence: If m n, then the horizontal asymptote will be.

4 2. Complete the table below. Graph the functions as needed to find the asymptotes. Function m n Leading coefficient of the numerator Leading coefficient of the denominator Equation of the horizontal asymptote Generalize your results by completing the sentence: If m n, then the horizontal asymptote is. (You may use words to describe the result or read section 2.6 for notation ideas.) 4. Slant asymptotes. Sometimes a rational function s end behavior is not defined by horizontal asymptotes but by a slant asymptote. All these functions have slant asymptotes. (Graph as many as you need to see what a slant asymptote looks like.) What is the relationship between m and n? What needs to be true about the relationship between m and n for a rational function to have a slant asymptote?

5 Rational Functions Worksheet: Due Based on the explorations, provide the partial analysis required for the following functions. Do so without a calculator. Then, verify your results with a calculator. Function Domain Zeros (xintercepts) Horizontal Asymptotes Vertical Asymptotes Type of end behavior: horizontal or slant asymptote (hint: factor out the 4 in the denominator, then factor the resulting trinomial) (Hint: remember all the ways we have to find the zeros of a polynomial!) 2. State the degree of the numerator and the degree of the denominator for each function. Then, state the horizontal asymptote OR why the function does not have a horizontal asymptote. Function m n Equation of horizontal asymptote OR Statement of why one does not exist

6 3. Match each graph on the next page with the correct equation below. Justify your answers by discussing vertical asymptotes, horizontal asymptotes, zeros, and end behavior. Equations:

7 Graphs: A D B E C F