Graphing Rational Functions

Transcription

1 5 LESSON Graphing Rational Functions Points of Discontinuit and Vertical Asmptotes UNDERSTAND The standard form of a rational function is f () 5 P(), where P () and Q () Q() are polnomial epressions. Remember that a parent function is the most basic in a famil of functions. The parent rational function is f () 5 _. A table of values and the graph for f () 5 _ are shown. f () vertical asmptote 0 UNDERSTAND The domain of a rational function f () 5 P () is the set of all real numbers Q () ecept those that make Q() 5 0. For an value of that makes the function undefined (i.e., that makes the denominator zero), there will be a point of discontinuit in the graph at that -value. At a point of discontinuit, there will either be a hole in the graph or a vertical asmptote, which is a vertical line that the graph approaches but never intersects. If P() and Q() have no common zeros, the graph of the function will have a vertical asmptote at each value of where Q() 5 0. If P() and Q() have a common zero, there is a hole in the graph at this value of, indicated b an open circle. Look at the parent function above. The numerator and denominator have no common zeros, and the value 5 0 is the onl point of discontinuit. There is a vertical asmptote at 5 0 (the -ais). As the graph approaches the asmptote from the left, it decreases to `. As it approaches from the right, it increases to `. Duplicating this page is prohibited b law. 0 Triumph Learning, LLC 0 Unit : Polnomial, Rational, and Radical Relationships

2 Connect Graph the rational function f () 5, and define its domain. Rewrite the function in factored form. The numerator is a difference of squares. ( )( ) f () 5 Find an points of discontinuit, and define the domain. 3 Graph the function. Remove the common factors to simplif the rational epression. ( )( ) f () 5 f () 5 The resulting function is linear. The graph is a line with a discontinuit at 5. When 5, the denominator is equal to 0. There is a point of discontinuit at 5. The domain of f () is all real numbers ecept. Duplicating this page is prohibited b law. 0 Triumph Learning, LLC 0 TRY Describe the graph of the function g() 5 3. Lesson 5: Graphing Rational Functions

3 UNDERSTAND Look again at the graph of f () 5 _. Now consider what happens when ver large numbers are substituted for. f (00) f (,000) 5, f (0,000) 5 0, Horizontal Asmptotes Note that as the value of increases, f () decreases, coming etremel close to zero. But it can never equal zero, no matter how great the value of becomes. This is shown on the graph as a horizontal asmptote at 5 0 (the -ais). As approaches both ` and `, f () approaches 0. As approaches `, the graph approaches zero from below. As approaches `, the graph approaches zero from above. 0 horizontal asmptote UNDERSTAND A rational function can have at most one horizontal asmptote. Use the following to determine whether and where the graph of a rational function f () 5 P() Q() has a horizontal asmptote. If the degree of P() is less than the degree of Q(), the horizontal asmptote is 5 0 (the -ais). If P() and Q() have the same degree, the horizontal asmptote is the line 5 a b, where a is the leading coefficient of P() and b is the leading coefficient of Q(). If the degree of P() is greater than the degree of Q (), there is no horizontal asmptote. In the case of the parent function, f () 5 _, the numerator has degree 0, the denominator has degree, so the horizontal asmptote is the -ais. With the function f () 5, the numerator has degree, the denominator has degree, and there is no horizontal asmptote. In that case, the function simplifies to a linear function. Duplicating this page is prohibited b law. 0 Triumph Learning, LLC Unit : Polnomial, Rational, and Radical Relationships

4 Connect 5 Graph the function f () 5 3. Determine if there is a horizontal asmptote. 3 The numerator and denominator have the same degree. In this case, both are linear polnomials. So, there is a horizontal asmptote. The leading coefficient of the numerator is. The leading coefficient of the denominator is. There is a horizontal asmptote at 5, or 5. Find a few points on the graph. Find an points of discontinuit. The denominator equals zero when 5 3. There are no common factors between the numerator and denominator. There is a vertical asmptote at 5 3. To find the -intercept, set the numerator equal to zero, and solve Sketch a graph. Duplicating this page is prohibited b law. 0 Triumph Learning, LLC DISCUSS To find the -intercept, find f (0). (0) The -intercept is ( 0, 5 3 ) Describe the behavior of the graph as it approaches the vertical asmptote from either side. How does the value of f () change as gets ver close to 3? Lesson 5: Graphing Rational Functions 3

5 EXAMPLE A Graph the rational function f () ( ) Write the rational epression in factored form, and simplif. Factor the numerator b grouping. Factor the denominator b inspection. ( 9)( ) f () 5 ( )( ) ( 3)( 3)( ) f () 5 ( )( ) Remove the common factor of to simplif. Rename the function. ( 3)( 3) g() 5 It is important to remember that there will be a hole in the graph at 5 because is a zero of both the numerator and denominator of f (). Find the point of discontinuit. g() The graph of f () will have a hole at (, 5 3 ). Find an vertical asmptotes. Set the denominator of the simplified function, g(), equal to 0, and solve There will be a vertical asmptote at 5. 3 Find a horizontal asmptote, if one eists. In f (), the degree of the numerator is greater than the degree of the denominator. The graph has no horizontal asmptote. Find the intercepts. To find the -intercepts, set the numerator of g() equal to 0, and solve. ( 3)( 3) and 5 3 The -intercepts are (3, 0) and (3, 0). To find the -intercept, find g(0). g(0) 5 (3)(3) 5 9 The -intercept is (0, 9). Duplicating this page is prohibited b law. 0 Triumph Learning, LLC Unit : Polnomial, Rational, and Radical Relationships

6 5 Make a table of values to find several more points Draw a graph. Use all of the information gathered. 0 TRY Duplicating this page is prohibited b law. 0 Triumph Learning, LLC The graph of g() is shown. 0 Identif the solutions of g() 5 0. Lesson 5: Graphing Rational Functions 5

7 EXAMPLE B Graph f () 5. Write the rational epression in factored form, and simplif. The denominator can be rewritten as. f () 5 ( )( ) f () 5 ( )( )( ) There are no common factors. The function cannot be simplified further. Find an vertical asmptotes. Set the denominator equal to 0, and solve. ( )( )( ) 5 0 The first factor has no real solutions. The other two factors produce real solutions: 5 and 5 There will be vertical asmptotes at 5 and 5. 3 Find a horizontal asmptote, if one eists. Note that the denominator is not written in standard form. The leading term of the polnomial in the denominator is. Both the numerator and denominator are th degree polnomials, so there is a horizontal asmptote. Find the quotient of the leading coefficients: 5. There is a horizontal asmptote at 5. Find the intercepts. To find the -intercepts, set the numerator equal to 0, and solve The -intercept is at (0, 0). To find the -intercept, find f (0). f (0) 5 (0 ) ( 0 ) The -intercept is (0, 0). Duplicating this page is prohibited b law. 0 Triumph Learning, LLC Unit : Polnomial, Rational, and Radical Relationships

8 5 Make a table of values to find several more points Draw a graph. Use all of the information gathered. 0 Duplicating this page is prohibited b law. 0 Triumph Learning, LLC TRY Graph the function g() Lesson 5: Graphing Rational Functions 7

9 Practice Find each attribute for the given function. If none eist, write none.. f () horizontal asmptote: vertical asmptote(s): (, ) coordinates of point(s) of discontinuit: REMEMBER Give asmptotes as equations. 0. f () 5 35 horizontal asmptote: vertical asmptote(s): (, ) coordinates of point(s) of discontinuit: HINT A common factor that is factored out of the denominator will leave a hole in the graph f () horizontal asmptote: vertical asmptote(s): (, ) coordinates of point(s) of discontinuit: REMEMBER Look at the leading terms of the numerator and denominator to determine horizontal asmptotes. For each graph, find all solutions to the equation f () Duplicating this page is prohibited b law. 0 Triumph Learning, LLC 8 Unit : Polnomial, Rational, and Radical Relationships

10 Choose the best answer.. The graph of f () is shown. Which of the following statements is not true? 7. The graph of g() is shown. Which of the following statements is not true? 0 0 A. The numerator and denominator share a common factor of ( ). B. The denominator has a factor of ( ). C. The degree of the numerator is the same as the degree of the denominator. D. There are no solutions to the equation f () 5 0. A. The degree of the numerator is the same as the degree of the denominator. B. The leading coefficient of the numerator is the same as the leading coefficient of the denominator. C. When 5, the numerator of the function equals 0. D. The numerator and denominator have no common factors. Sketch a graph of each function. Duplicating this page is prohibited b law. 0 Triumph Learning, LLC 8. f () g() Lesson 5: Graphing Rational Functions 9

11 Solve. 0. The graph of f () is shown. 0 3 Compare the graph of f () to the graph of g() 5.. Several co-workers carpool to work each da. The dail cost of gas and tolls, $, is split evenl among the people who carpool that da. The cost per person varies inversel with the number of people. Write a function that describes the cost per person on a given da if people carpool. c() 5 Graph the function Which parts of the graph make sense in contet? Eplain. Duplicating this page is prohibited b law. 0 Triumph Learning, LLC 30 Unit : Polnomial, Rational, and Radical Relationships

12 Solve.. EXTEND Solve: 5 Write a function for each side of the equation, and graph both functions on the coordinate plane below. a() 5 b() 5 0 The points of intersection are (, ) and (, ). Remember that the original equation has onl variable. The solution is and. Duplicating this page is prohibited b law. 0 Triumph Learning, LLC 3 3. EXTEND Graph the inequalit:. 3 Graph the related equation 5. Recall that when graphing an inequalit in two variables, portions of the graph are shaded. In the graph of a rational inequalit, the shaded sections are separated b the graph s curves and the asmptotes. 0 Lesson 5: Graphing Rational Functions 3