lesson-2-reperentation-of-rational-function (1).pptx

Lesson 2 Representing Rational Functions through Tables, Graphs, and Equations  Objectives At the end of this lesson, the learner should be able to correctly construct table of values for rational functions; correctly determine the asymptotes of rational functions; and accurately draw t...

Lesson 2 Representing Rational Functions through Tables, Graphs, and Equations  Objectives At the end of this lesson, the learner should be able to correctly construct table of values for rational functions; correctly determine the asymptotes of rational functions; and accurately draw the graphs of rational functions.  Essential Questions How will you express rational functions using a table of values? How will you determine the asymptotes of rational functions? How will you graph a rational function?  Warm Up! For this lesson, we will discuss representing rational functions through tables, graphs, and equations. Before we proceed, let us read this comic strip to have an idea about asymptotes, one core concept in this lesson. (Click on the link below to view the comic.) Memes Blog. (2012). Memes Blog: Asymptote-zoned. [online] Available at: https://bit.ly/2VzSEow [Accessed 8 April 2019]. Asymptote-zoned Can you relate? Asymptote-zoned Can you relate? Asymptote-zoned Can you relate?  Guide Questions What do you think is an asymptote based on the comic strip? Do you think being in a “friendzone” is similar to the definition of an asymptote? What do you think are the different ways on how can we represent a function?  Learn about It! Table of values 1 composed of values and or that satisfy the given function Examples: Table of values for undefine d  Learn about It! Table of values 1 composed of values and or that satisfy the given function Examples: Table of values for undefine d  Learn about It! Asymptote 2 a line that a curve approaches but does not intersect. Example: The line is an asymptote of the graph on the right.  Learn about It! Vertical Asymptote 3 a vertical line of the form which the curve approaches but never touches. The vertical line passing through the zeroes of the denominator of the rational functions are the vertical asymptotes. Examples: The vertical asymptote of the function is the line.  Learn about It! Vertical Asymptote 3 a vertical line of the form which the curve approaches but never touches. The vertical line passing through the zeroes of the denominator of the rational functions are the vertical asymptotes. Examples: The vertical asymptote of the function is the line.  Learn about It! Horizontal Asymptote 4 a horizontal line of the form which the curve approaches but never touches To determine the horizontal asymptote of a rational function , the degrees of the numerator and the denominator will be considered. Let and be the degree of and respectively. a. If , the horizontal asymptote of is the line. b. If , he horizontal asymptote is , where is the leading coefficient of and is the leading coefficient of. c. If , there is no horizontal asymptote. Examples: The rational function has no horizontal asymptote.  Learn about It! Horizontal Asymptote 4 a horizontal line of the form which the curve approaches but never touches To determine the horizontal asymptote of a rational function , the degrees of the numerator and the denominator will be considered. Let and be the degree of and respectively. a. If , the horizontal asymptote of is the line. b. If , he horizontal asymptote is , where is the leading coefficient of and is the leading coefficient of. c. If , there is no horizontal asymptote. Examples: The rational function has its horizontal asymptote at.  Learn about It! Horizontal Asymptote 4 a horizontal line of the form which the curve approaches but never touches To determine the horizontal asymptote of a rational function , the degrees of the numerator and the denominator will be considered. Let and be the degree of and respectively. a. If , the horizontal asymptote of is the line. b. If , he horizontal asymptote is , where is the leading coefficient of and is the leading coefficient of. c. If , there is no horizontal asymptote. Examples: The horizontal asymptote of the function is.  Learn about It! Oblique Asymptote 5 a line of the form which the curve approaches but never touches. Oblique asymptotes exist in a function if the degree of the numerator is larger than the degree of the denominator , that is,. Example: The rational function has an oblique asymptote. If we divide , the quotient is. Thus, the equation of the oblique asymptote is.  Try It! Example 1: Construct a table of values for the rational function.  Try It! Example 1: Construct a table of values for the rational function. Solution: Assign values for. Observe that the denominator of is. When , then the function becomes undefined. We can, instead, select those values less than zero and more than zero in our table of values.  Try It! Example 1: Construct a table of values for the rational function. Solution: Note that we are free to select whatever values of we want to have. However, it would be useful to select those values close to the value of that makes the function undefined.  Try It! Example 1: Construct a table of values for the rational function. Solution: Substitute the values of to the function to get the corresponding values of. Example, for ,.  Try It! Example 1: Construct a table of values for the rational function. Solution: Continue to substitute the values of to the function to get the corresponding values of.  Try It! Example 2: Construct a table of values for the rational function. Then, determine the vertical and horizontal asymptotes of the function.  Try It! Example 2: Construct a table of values for the rational function. Then, determine the vertical and horizontal asymptotes of the function. Solution: Assign values for. Observe that , will make the function becomes. Instead, select those integers less than 1 and more than 1.  Try It! Example 2: Construct a table of values for the rational function. Then, determine the vertical and horizontal asymptotes of the function. Solution: Substitute the values of to the function to get the corresponding values of.  Try It! Example 2: Construct a table of values for the rational function. Then, determine the vertical and horizontal asymptotes of the function. Solution: Determine the asymptotes. The rational function is undefined at It means the graph of the function will not intersect with the line. Hence, the vertical asymptote is the line.  Try It! Example 2: Construct a table of values for the rational function. Then, determine the vertical and horizontal asymptotes of the function. Solution: To determine the horizontal asymptote, we have to determine the degrees of the numerator and the denominator. The degree of the numerator is while the degree of the denominator is. Thus, the case is. Thus, it means that the horizontal asymptote is.  Try It! Example 2: Construct a table of values for the rational function. Then, determine the vertical and horizontal asymptotes of the function. Solution: Thus, the vertical and horizontal asymptotes of are and respectively. Its table of values is presented below.  Let’s Practice! Individual Practice: 1. Construct a table of values for the rational function. 2. Determine the vertical and horizontal asymptotes of  Let’s Practice! Group Practice: To be done in 2 to 5 groups Sketch the graph of. Key Points Table of values 1 composed of values and or that satisfy the given function Asymptote 2 a line that a curve approaches but does not intersect. Vertical Asymptote 3 a vertical line of the form which the curve approaches but never touches. The vertical line passing through the zeroes of the denominator of the rational functions are the vertical asymptotes. Key Points Horizontal Asymptote 4 a horizontal line of the form which the curve approaches but never touches To determine the horizontal asymptote of a rational function , the degrees of the numerator and the denominator will be considered. Let and be the degree of and respectively. a. If , the horizontal asymptote of is the line. b. If , he horizontal asymptote is , where is the leading coefficient of and is the leading coefficient of. c. If , there is no horizontal asymptote. Oblique Asymptote 5 a line of the form which the curve approaches but never touches. Oblique asymptotes exist in a function if the degree of the numerator is larger than the degree of the denominator , that is,.  Synthesis What are the different ways that we represent a rational function? Why is it necessary to represent rational functions in different ways? In a rational function, is it correct to say that there is always a corresponding value of for any given value of ?