Unit 2: Rational Functions,Lesson 1: Reciprocal of a Linear Function,What is a Rational Function?,Any function of the form:Where f(x) and g(x) are polynomial functionsBecause the denominator can never be zero, rational functions have properties that polynomial functions do not,What is the Reciprocal of a Linear Function?,We are going to start by looking at the simplest rational functions:This is the general form for the reciprocal of a linear function Reciprocal means you put “one over” or, more simply, you “flip it”,Example 1,(a) Use your TI-83 or the program “Graph” to graph the function(b) Describe the end behaviour (c) What happens when x gets close to ?,Example 1: Solution,(a),Example 1: Solution,(b) As x gets large in both the positive and negative directions, the function gets close to but does not touch the y-axis. Therefore,As x +, y 0As x , y 0,Example 1: Solution,Start on the left of the graph and move towards x = : the y-values get large and negative,Start on the right of the graph and move towards x = : the y-values get large and positive,Denoted by: x - , y -,Denoted by: x + , y +,Approach x = from the right,Approach x = from the left,(c),The function never crosses this vertical line,Example 1: Notes,A line that a function gets close to but does not touch is called an asymptoteThe y-values got close to but did not touch the y-axis (a horizontal line) horizontal asymptote is the line y = 0 The reciprocal of a linear function will always have a horizontal asymptote at y = 0The x-values got close to but did not touch the line x = (a vertical line) vertical asymptote is the line x = Occurs because the denominator cannot be zero The reciprocal of a linear function will always have one vertical asymptote,Example 2,(a) Use your TI-83 or the program “Graph” to graph the function(b) Label the horizontal and vertical asymptotes,Example 2: Solution,The line x = 1 (vertical asymptote),The line y = 0 (horizontal asymptote),Example 2: Notes,In this example the value of k (the number in front of x) is negative:As a result, branch on the left of the vertical asymptote is above the x-axis and the branch on the right branch is below itWhen k is positive the branch on the left is below the x-axis and the branch on the right branch is above it,Example 3,Consider the function(a) Determine the equations of the asymptotes (b) State the domain and range,Example 3: Solution,The horizontal asymptote is the line y = 0 See “Example 1: Notes”The vertical asymptote occurs because the denominator cannot be zero. We need to find the value of x that makes the denominator zeroTherefore, the vertical asymptote is the line x = 2,Example 3: Solution,(b) The domain tells us what values of x the function can be evaluated at. The only value of x we cant have is 2. Therefore, The range tells us what values of y the function can have. The only value of y we will never get is 0. Therefore,Example 3: Notes,Our vertical asymptote was the line x = 2 and our domain was The vertical asymptote gives you the domainOur horizontal asymptote was the line y = 0 and our range was The horizontal asymptote gives you the range,Example 4,Determine the x- and y-intercepts of,Example 4: Solution,The x-intercept is the value of x when y = 0:There is no value of x that makes this true. There is no x-interceptThe y-intercept is the value of y when x = 0:The y-intercept is,Summary,The reciprocal of a linear function has the formThe vertical asymptote is found by setting the denominator equal to zero and solving for x The denominator CANNOT be zero The domain is all values of x except this oneThe horizontal asymptote is the x-axis (the line y = 0) The range is all values of y except zeroThese functions have two branches one on the left of the vertical asymptote and one on the right k 0: left branch is below the x-axis, the right is above K 0: left branch is above the x-axis, the right is below,Practice Problems,P. 153-154 #2, 3, 5, 7-9 Note: For #7 dont bother with a sketch. Just calculate the y-intercept and state the domain, range and asymptotes.,