7h2 Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)

Transcription

1 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Goals: Graph rational functions Find the zeros and y-intercepts Identify and understand the asymptotes Match graphs to their equations Vocabulary: Function notation: Domain: Asymptote: Intercepts: Zero: Table of values: Graph: Sketch: Function notation: Function f with input can be written in several ways: f() = equation rule f: equation rule f: f()= equation rule In all cases, when you input a value for, the output is the y coordinate. E. (, y) is the same as (, f() ) on the ais on the y ai E. A. For each of the following: a. find f(0), f(-), f() b. Graph the function y = f () i. f() = + ii. f :

2 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Graphing Simple Broken Rational Functions Definition of a Rational Function: A Rational Function is a function that has a variable,, in the denominator. For eample: f : f () = 5 f : + g : g() = + They all have a restriction on. Therefore they have a restriction on their domain. That is why we call them broken. E. B a. Fill the table of values for = to = for g : + y b. Graph the function g At the break there is special behaviour. Let s check out what happens when is close to, i.e. =.97,.98,.99,.00,.0,.0, = y error! broken This special behaviour is what defines an Asymptote.

3 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Asymptotes An asymptote is a line which a graph comes closer and closer to, but never crosses. Eploring Asymptotes: Activity: Looking at the graph of f ( ) = we notice? using online graphing software (e. geogebra), what do Eploring graphing the broken rational function: E.C. Fill in the following table of values for each function. Plot the points, and graph the function. a. f : y

4 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) b. g : y c. h : y

5 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Vertical Asymptotes: Finding the Asymptotes Vertical asymptotes occur at the restrictions on the domain. i.e. at the restriction on the in the denominator. When N, the equation of the vertical asymptote is = N. Horizontal Asymptotes: The horizontal asymptote is a value that the output, y, will never reach, but will come closer and closer to. Here are two methods to find the horizontal asymptote. or Method : substitute in very large positive and negative values for (etreme values) and see where the y value tends towards (comes closer to) i.e. f(-0 000) f(-000) f(000) f(0 000) Method : divide every term by. Eliminate whatever will become insignificant as becomes infinitely big. Think: for eample: becomes insignificant as because is very very small. Also true for 5 ; as, 5 0 or any number: number ; as, number 0 Worked Eample: Find the horizontal asymptote for f ( ) = by Method : f ( 000) = f ( 000) = = by Method : = ( 000) = = ( ) = = as, 0 so close to 0.5 or 0 ( ) = the horizontal asymptote occurs at y = - - y = 5

6 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) E. D Find the horizontal asymptotes. At first, use both methods. Then choose one method that you prefer. a. f : + b. g() = c. f ( ) = + d. h : + e. g : g( ) = f. h( ) = + ( ) Graphing Rational Functions E. E Graph the first three functions from E. D. (first find and mark the asymptotes, then create a table of values). a. f : + b. g() = c. f ( ) = + 6

7 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) The Shape of Things (Remember: graphs hug asymptotes) How to sketch a graph given:. The and y intercepts (if there are any). The vertical asymptote(s). The horizontal asymptote E. Sketch the graph with the given parameters: Using the shape of things to sketch graphs of rational functions (no tables of values). Find and mark the vertical asymptote on the graph Vertical asymptotes occur at the restriction on in the denominator.. Find and mark the horizontal asymptote on the graph Use either method or to find the horizontal asymptote (ie. f(000) & f(-000) or divide by and eliminate insignificants). Find and y intercepts and mark them on the graph. for y intercepts: let = 0 find y. ( 0, ) for intercepts: let y = 0 find. (, 0 ) 7

8 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) E.F Graph each of the following using the sketching method: a. f ( ) = + b. f : 8

9 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Match the Graph Look for key features: Intercepts and Asymptotes E. G Match the graph to its equation: a. y = b. f () = c. f : ( ) d. f : 9

10 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Noticing Patterns: Use graphing software to graph each set of functions on the same set of ais and then discuss the differences and similarities: A B C D y = y = + f ( ) = f : ( +) y = 6 y = f ( ) = f : y = y = + f ( ) = f : 6 + y = 0. y = + f ( ) = Practice Eercises: P08 Eercises Mied (salad of) solutions for : Which number(s) cannot be included in the domain? Indicate a domain for each one. a) f : b) f :() = c) f : d) f : 9 e) f : () = z f) f : ( ) ( ) g) t f : h) f : t 5 t t i) s f : k) f : s For the following functions you cannot calculate the restrictions on the domain directly. Why not? Try to find the numbers using systematic sampling. a) b) c), 5 f : g : 8 7 h : 8 0

11 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) P09 Use a function plotter to draw the graphs of the functions. Indicate the equations of horizontal and vertical asymptotes, if they eist. Describe how you could also have found these without th graph the graph by by using the the function rule. (equation). a) f : b) f : c) f : d) f : s, 5 s t e) f : t f) f :, 5 t ( ) ( ) 5 Given each function rule, consider ansider if the the graph includes asymptotes and, if relevant, indicate the equations of the asymptotes. Draw the function graph (without using a table of values or a function plotter). a) f : b) g : ( ), 5 c) h :( ) = d) k : e) p : f) q : 0, 5 ( ) 9 Which numbers must be ecluded from the definition set? Indicate the definition set and sketch the function graphs. a) f : b) g : ( ) ( ) 0 For Indicate each, write each the time rule a for broken a broken rational function which has the line with the given equation as asymptote. a) = b) y = - c) = 0 ; the graph should also go through the point ( ). Indicate a broken rational function for which the graph does not enter the yellow area. (Write the function rule for a graph). 7 a) Draw the graph of the function f :. Describe how the graph would change if the number was replaced with - a larger number - a smaller number b) What would change in the graph of f if in the numerator the variable was included instead of the number? c) Using a function plotter draw the graphs of the functions f : ; f : and f : in a common coordinate system and describe the differences between the graphs.

12 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) P0 The function graphs shown should be matched to the function terms rules given. In the order a) to h) the correctly matched letters result in a solution word. But first you have to >break> another code. Indicate the a broken rational functions for which the graphs intersection go through of asymptotes the point occurs P ( ). at the point P( ) Draw the graph of the function f :, How would the graph change if the was replaced by a larger number? By a smaller number? 5 The following functions each have a restriction on the domain. The graphs of the function, however, show four differenz t behaviours near these restrictions. Draw the function geaphs with a function plotter and describe the four different behaviours. a) f : b) f : c) f : d) f : e) f 5 : f) f 6 : ( ), 5 g) f 7 : h) f 8 : t Could you determine which of these behaviours will occur just by considering the function rule? P6 : f : ( ) C f : W f : ( ) S f M : f J f : B ( 6)( ) ( ) f : B f : ( ) F Find the intersections of the graphs of f : and g : mathematically. Sketch both function graphs on the same aes.

13 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) Etra Practice Graphing: Graph each of the following functions. Indicate the intercepts and equations for the asymptotes in each case. a. y = d. f ( ) = + g. f : + b. y = e. f ( ) = + h. f : + c. f ( ) = f. f ( ) = + i. h :h() = ( ) Answers: S.08 S.09

14 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen)

15 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) S.0 5

16 7h Graphing Basic Rational Functions (Gebrochenrationaler Funktionen) S.6 Etra Practice Graphing: Answers: a. b. c. d. e. f. g. h. i. 6