. Graphs of Reciprocal Functions YOU WILL NEED graph paper coloured pencils or pens graphing calculator or graphing software f() = GOAL Sketch the graphs of reciprocals of linear and quadratic functions. INVESTIGATE the Math Owen has noted some connections between the graphs of f () and its reciprocal function g(). Both graphs are in the same quadrants for the same -values. When f (), there is a vertical asmptote for g(). f () is alwas increasing, and g() is alwas decreasing. g() =? How are the graphs of a function and its reciprocal function related? A. Eplain wh the graphs of f () and g() are in the same quadrants over the same intervals. Does this relationship hold for m() and n ()? Does this relationship hold for an function and its reciprocal function? Eplain. B. What graphical characteristic in the reciprocal function do the zeros of the original function correspond to? Eplain. C. Eplain wh the reciprocal function g() is decreasing when f () is increasing. Does this relationship hold for n () and m()? Eplain how the increasing and decreasing intervals of a function and its reciprocal are related. D. What are the -coordinates of the points where f () and g() intersect? Will the points of intersection for an function and its reciprocal alwas have the same -coordinates? Eplain. E. Eplain wh the graph of g() has a horizontal asmptote. What is the equation of this asmptote? Will all reciprocal functions have the same horizontal asmptote? Eplain. 8. Graphs of Reciprocal Functions NEL
F. On graph paper, draw the graph of p(). In a table like the one below, note the characteristics of the graph of p() and use this information to help ou determine the characteristics of the reciprocal function q().. Characteristics p() q() zeros and/or vertical asmptotes interval(s) on which the graph is above the -ais (all values of the function are positive) interval(s) on which the graph is below the -ais (all values of the function are negative) interval(s) on which the function is increasing interval(s) on which the function is decreasing point(s) where the -value is point(s) where the -value is G. On the same graph, draw the vertical asmptotes for the reciprocal function. Then use the rest of the information determined in part F to draw the graph for q(). H. Verif our graphs b entering p() and q() in a graphing calculator using the friendl window setting shown. I. Repeat parts F to H for the following pairs of functions. a) p() and q() b) p() 3 and q() 3 c) p() ( )( 3) and q() d) p() ( ) and q() ( ) ( )( 3) J. Write a summar of the relationships between the characteristics of the graphs of a) a linear function and its reciprocal function b) a quadratic function and its reciprocal function Tech Support On a graphing calculator, the length of the displa screen contains 9 piels, and the width contains piels. When the domain, Xma-Xmin, is cleanl divisible b 9, and the range, Yma-Ymin, is cleanl divisible b, the window is friendl. This means that ou can trace without using ugl decimals. A friendl window is useful when working with rational functions. Use brackets when entering reciprocal functions in the Y editor of a graphing calculator. For eample, to graph the function f(), enter Y. ( ) NEL Chapter 9
Reflecting K. How did knowing the positive/negative intervals and the increasing/decreasing intervals for p() help ou draw the graph for p()? L. Wh are some numbers in the domain of a function ecluded from the domain of its reciprocal function? What graphical characteristic of the reciprocal function occurs at these values? M. What common characteristics are shared b all reciprocals of linear and quadratic functions? APPLY the Math EXAMPLE Connecting the characteristics of a linear function to its corresponding reciprocal function Given the function f (), a) determine the domain and range, intercepts, positive/negative intervals, and increasing/decreasing intervals b) use our answers for part a) to sketch the graph of the reciprocal function Solution a) f () is a linear function. D PR R PR From the equation, the -intercept is. f () when The -intercept is. The domain and range of most linear functions are the set of real numbers. A linear function f() m b has -intercept b. The -intercept occurs where f(). = f () is positive when P(`, ) and negative when P(, `). f () is decreasing when P(`, `). Sketch the graph of f() to determine the positive and negative intervals. The line is above the -ais for all -values less than and below the -ais for all -values greater than. This is a linear function with a negative slope, so it is decreasing over its entire domain.. Graphs of Reciprocal Functions NEL
b) The reciprocal function is g(). D PR R PR The -intercept is.. The vertical asmptote is and the horizontal asmptote is. The reciprocal function is positive when P(`, ) and negative when P(, `). It is increasing when P(`, ) and when P(, `). The graph of g() intersects the graph of g() at (, ) and (3, ). All the -values of points on the reciprocal function are reciprocals of the -values on the original function. There is a vertical asmptote at the zero of the original function. The reciprocals of a linear function alwas have the -ais as a horizontal asmptote. The positive/negative intervals are alwas the same for both functions. Because the original function is alwas decreasing, the reciprocal function is alwas increasing. The reciprocal of is, and the reciprocal of is. Thus, the two graphs intersect at an points with these -values.. = = = Use all this information to sketch the graph of the reciprocal function. EXAMPLE Connecting the characteristics of a quadratic function to its corresponding reciprocal function Given the function f () 9 a) determine the domain and range, intercepts, positive/negative intervals, and increasing/decreasing intervals b) use our answers for part a) to sketch the graph of the reciprocal function Solution a) f () 9 is a quadratic function. D PR The domain of a quadratic function is the set of real numbers. f () 9, so the -intercept is 9. R PR # 9 The graph of f() is a parabola that opens down. The verte is at (, 9), so # 9. NEL Chapter
f () 9 (3 )(3 ) 3 The -intercepts are 3 and 3. = 9 Factor and determine the -intercepts. The parabola 9 is above the -ais for -values between 3 and 3. The graph is below the -ais for -values less than 3 and for -values greater than 3. f () is positive when P(3, 3) and negative when P(`, 3) and when P(3, `). f () is increasing when P(`, ) and decreasing when P(, `). b) The reciprocal function is g(). 9 The vertical asmptotes are 3and 3. D PR 3 The horizontal asmptote is. The -intercept is. 9 There is a local minimum value at a,. 9 b R UPR, or $ 9 V The -values increase as increases from ` to. The -values decrease as increases from to `. Vertical asmptotes occur at each zero of the original function, so these numbers must be ecluded from the domain. The reciprocals of all quadratic functions have the -ais as a horizontal asmptote. The -intercept of the original function is 9, so the -intercept of the reciprocal function is. 9 When the original function has a local maimum point, the reciprocal function has a corresponding local minimum point.. Graphs of Reciprocal Functions NEL
The reciprocal function is positive when P(3, 3) and negative when P(`, 3) and when P(3, `). It is decreasing when P(`, 3) and when P(3, ), and increasing when P(, 3) and when P(3, `). f () when 9 and f () when 9 9 9 8 8 " " The positive/negative intervals are alwas the same for both functions. Where the original function is decreasing, ecluding the zeros, the reciprocal function is increasing (and vice versa). A function and its reciprocal intersect at points where. Solve the corresponding equations to determine the -coordinates of the points of intersection.. The graph of g() intersects the graph of f () 9 9 at (", ), (", ) and at (", ), (", ). = 9 = 9 Use all this information to sketch the graph of the reciprocal function. = 3 = 3 In Summar Ke Idea You can use ke characteristics of the graph of a linear or quadratic function to graph the related reciprocal function. Need to Know All the -coordinates of a reciprocal function are the reciprocals of the -coordinates of the original function. The graph of a reciprocal function has a vertical asmptote at each zero of the original function. A reciprocal function will alwas have as a horizontal asmptote if the original function is linear or quadratic. A reciprocal function has the same positive/negative intervals as the original function. (continued) NEL Chapter 3
Intervals of increase on the original function are intervals of decrease on the reciprocal function. Intervals of decrease on the original function are intervals of increase on the reciprocal function. If the range of the original function includes and/or, the reciprocal function will intersect the original function at a point (or points) where the -coordinate is or. If the original function has a local minimum point, the reciprocal function will have a local maimum point at the same -value (and vice versa). A linear function and its reciprocal = + = + A quadratic function and its reciprocal = (+) = (+) = = 3 = Both functions are negative when P(`, ) and positive when P(, `). The original function is increasing when P(`, `). The reciprocal function is decreasing when P(`, ) or (, `). Both functions are negative when P(3, ) and positive when P(`, 3) or (, `). The original function is decreasing when P(`, ) and increasing when P(, `). The reciprocal function is increasing when P(`, 3) or (3, ) and decreasing when P(, ) or (, `). CHECK Your Understanding. Match each function with its equation on the net page. Then identif which function pairs are reciprocals. a) c) e) b) d) f). Graphs of Reciprocal Functions NEL
A B C. For each pair of functions, determine where the zeros of the original function occur and state the equations of the vertical asmptotes of the reciprocal function, if possible. a) b) c) d) e) f) ( ) f (), g() f () 3, g() 3 f (), g() f (), g() f (), g() f () 3, g() 3 D E F ( ). 3. Sketch the graph of each function. Use our graph to help ou sketch the graph of the reciprocal function. a) f () b) f () PRACTISING. a) Cop and complete the following table. 3 3 7 f() 8 f() b) Sketch the graphs of f () and. f () c) Find equations for f () and. f (). State the equation of the reciprocal of each function, and determine the equations of the vertical asmptotes of the reciprocal. Verif our results using graphing technolog. a) f () e) f () 3 b) f () f) f () ( 3) c) f () g) f () 3 d) f () h) f () 3 NEL Chapter
. Sketch the graph of the reciprocal of each function. a) c) = f () = f () b) d) = f () = f () 7. Sketch each pair of graphs on the same aes. State the domain and range of each reciprocal function. a), b) 3, 3 8. Draw the graph of f () and on the same aes. f () a) f () d) f () ( 3) b) f () ( ) 3 e) f () c) f () 3 f) f () ( ) 9. For each function, determine the domain and range, intercepts, K positive/negative intervals, and increasing/decreasing intervals. State the equation of the reciprocal function. Then sketch the graphs of the original and reciprocal functions on the same aes. a) f () 8 c) f () b) f () 3 d) f (). Wh do the graphs of reciprocals of linear functions alwas have vertical asmptotes, but the graphs of reciprocals of quadratic functions sometimes do not? Provide sketches of three different reciprocal functions to illustrate our answer.. Graphs of Reciprocal Functions NEL
. k. An equation of the form has a graph that closel matches b c the graph shown. Find the equation. Check our answer using graphing technolog.. A chemical compan is testing the effectiveness of a new cleaning A solution for killing bacteria. The test involves introducing the solution into a sample that contains approimatel bacteria. The number of bacteria remaining, b(t), over time, t, in seconds is given b the equation b(t). t a) How man bacteria will be left after s? b) After how man seconds will onl bacteria be left? c) After how man seconds will onl one bacterium be left? d) This model is not alwas accurate. Determine what sort of inaccuracies this model might have. Assume that the solution was introduced at t. e) Based on these inaccuracies, what should the domain and range of the equation be? = = 3. Use our graphing calculator to eplore and then describe the ke T characteristics of the famil of reciprocal functions of the form g(). Make sure that ou include graphs to support our n descriptions. a) State the domain and range of g(). b) For the famil of functions f () n, the -intercept changes as the value of n changes. Describe how the -intercept changes and how this affects g(). c) If graphed, at what point would the two graphs f () and g() intersect?. Due to a basketball tournament, our friend has missed this class. C Write a concise eplanation of the steps needed to graph a reciprocal function using the graph of the original function (without using graphing technolog). Use an eample, and eplain the reason for each step. Etending. Sketch the graphs of the following reciprocal functions. a) c) " b) d) 3 sin. Determine the equation of the function in the graph shown. NEL Chapter 7