Introduction to Rational Functions

Introduction to Rational Functions The net class of functions that we will investigate is the rational functions. We will eplore the following ideas: Definition of rational function. The basic (untransformed) rational function. Table of and y values. Rational function and its inverse (algebraic and graphical eamples). How does y change as (positive) gets very large and very small? Symmetry of the basic rational function. Graph of the basic rational function. Domain (and restrictions) and range of the basic rational function. Notion of a limit. Let s first recall: what kind of numbers are rational numbers? Definition of Rational Function Just as we saw how the Division Algorithm for integers applies to polynomials or functions, the definition of rational numbers can be etended to functions. A rational function is any function of the form r numerator and denominator polynomials. polynomials. = n ( ) d, where n ( ) and d represent Rational functions then are RATIOS of The Basic Rational Function = The basic (untransformed) rational function is f. Complete the following table of values for y if = { 0.000, 0.00, 0.0,0., 0,,0,00,000,0000}. 0000 000 00 0 0. 0.0 0.00 0.000 0 y = Raelene Dufresne 2004 Rational Functions of 4

Questions: Use the table of values to answer the following questions.. Eplain in words: a) What does the function f do to in order to get y? b) What is f!, the inverse of f? (What must be done to the original y value to return to the original value?) 2. What do you notice about f and f!? [ ]? a) What is the result of the composition f! f [ ]? b) What is the result of the composition f f! c) What algebraic statement means that a function is its own inverse? 3. The statement! + is read approaches infinity. To approach a value means to get close to the value but not necessarily equal to (or on) the value. What (single) value does y approach as! +? In calculus terms, this is said as evaluate the limit as! + of y. In calculus notation, this is written as: Evaluate lim y (or evaluate lim!+!+ ). 4. The statement! 0 is read approaches zero. From the table of positive values, what value does y approach as! 0? 5. How does the sign of y relate to the sign of? Determine the symmetry of f 6. Sketch the graph of f( ) =. 7. By looking at the equation f( ) =, determine the domain. 8. By looking at the graph of f( ) =, determine the range. 9. Reflect f( ) = =. through the line y =. What do you notice about the result? How does this relate to your answer in #2? 0. Use the graph of f( ) = to evaluate the following limits: a) lim!2 b) lim! 4 3 c) lim!# d) lim!0 e) lim!0 + f) lim!0 Raelene Dufresne 2004 Rational Functions 2 of 4

Elements of Rational Functions Consider the four different magnifications of the graph of f( ) = given below. Notice that the graphs don t cross either ais; therefore f( ) = does not have or y intercepts. as gets very small (close to zero), y gets very big (y approaches positive or negative infinity) as gets very big (as approaches positive or negative infinity), y gets very small (close to zero). f (! ) =!f( ); therefore f has odd symmetry. = The basic rational function y = may be considered the Reciprocal Function, since every output y value is the reciprocal of the input value (and vice versa). Asymptotic Behaviour: The behaviour of the y value of graphs to reach bounded values (limits) or to grow unboundedly (to approach infinity) is called Asymptotic Behaviour. Asymptotes are boundary lines (or curves) that act as limiters or attractors of the shape of the graph. Some types of asymptotes may be crossed by the graph and while other types of asymptotes will never be crossed by the graph. Elements of graphs are pieces of information that enables us to sketch the graph. Asymptotes are eamples of elements of rational functions. Other eamples are intercepts and holes in the graph. ELEMENTS OF RATIONAL FUNCTIONS. Holes in the Graph: If the numerator and denominator have a common factor, then the rational function will have a hole in the graph. (There will be the same number of holes in the graph as there are common factors.) a) A hole in the graph is really the absence of a point on the graph at a particular set of coordinates. Raelene Dufresne 2004 Rational Functions 3 of 4

b) Fully factor every rational function to test for holes in the graph. If there are common factors, reduce the fraction (the rational function). Then, determine the root associated with the common factor and rewrite the rational function as follows: Eample: r( ) = 2! 2 + 3 + 2 = + + 2 (!) ( + r ( ) =! + 2, #! c) The coordinate of the hole is the root of the common factor: hole =!. d) The y coordinate of the hole is result of the value substituted into the = reduced rational function: y hole = r!!!! + 2 =!2 =!2., where = c is the e) In calculus, we write the y coordinate of the hole as limf!c root of the factor common to the numerator and the denominator. Therefore, for this eample, the y-coordinate of the hole is identified by the limit lim! 2 2 + 3 + 2 = lim! ( ) ( + ) uuuuuuuu ( + 2) + = lim! ( ) ( + 2) = + 2 = 2., which is to say that for the f) Therefore the coordinates of the hole are!,!2 graph, (, y)! (, 2). To remove this point from the graph (or to plot a hole ), sketch a small circle at the coordinates of the hole and sketch the rest of the graph as usual. Eercise: Determine the simplified form of the equation of the rational function y = 2 + 6 2! 4! 2 and the coordinates of the hole. 2. Intercepts of (any) function: The intercepts of a function are the points on a graph that intersect the - or y aes. a) To determine the intercept(s): Set the y coordinate equal to zero and solve for. This results in setting the numerator function of a rational function equal to 0 and solving for. There may be one or several intercepts. b) To determine the y intercept: Set the coordinate equal to zero and solve for y. As is true for ALL functions, rational functions have ONLY ONE y intercept. c) Rational functions may or may not have intercepts. For eample, y = does NOT have any intercepts while y = + 2! has both and y intercepts. Raelene Dufresne 2004 Rational Functions 4 of 4

Eercise: Determine the intercepts of y = 2! 4!. 3. Vertical Asymptote: What value(s) of make(s) y get very large? a) A vertical asymptote of a (rational) function is the vertical line = h such that as! h, y! ±. b) There may be ONE or MORE THAN ONE VA (vertical asymptote) for rational functions. c) The reciprocal of infinity is zero. Therefore, in order to make the y value infinity, the denominator of the rational function must be zero. d) To determine the equation of the vertical asymptote(s), first ensure that you have checked for common factors and reduced the rational function. Then, set the denominator of a rational function equal to zero d State VAs as equations of vertical lines. e) The graph of a rational function NEVER crosses a vertical asymptote. Eercise: Determine the equation(s) of the VA of y = 2 +! 6. 4. Non-Vertical Asympototes: What happens to y as gets very large? a) Non-vertical asymptotes have equations of the form y = Q = 0 and solve for., where Q( ) represents the quotient of the numerator divided by the denominator. b) There will be ONLY ONE non-vertical asymptote for rational functions. c) To determine the equation of the non-vertical asymptote, first ensure that you have checked for common factors and reduced the rational function. Then, set y equal to the quotient of the rational function s numerator divided by its denominator. Therefore y = Q AS LONG AS R! 0. is the equation of the non-vertical asymptote d) The graph of a rational function MAY cross a non-vertical asymptote, but it does not have to. e) The order of the quotient (and therefore the shape of the non-vertical asymptote) depends on the orders of the numerator and denominator. Let s and d : has the form y = k, where k!r. is, then there is a HA @ y = 0, then there is a HA @ y = k, k! 0. investigate three possible cases for the orders of n Case I/ Horizontal Asymptote: y = Q If there are no common factors and the order of n LESS THAN the order of d EQUAL TO the order of d Raelene Dufresne 2004 Rational Functions 5 of 4

A horizontal asymptote of a (rational) function is the horizontal line y = k such that as! ±, y! k. Case II/ Slant Asymptote: y = Q The order of n has the form y = m + b, where b, m!r. is ONE GREATER THAN the order of d (as long as there are no common factors). A slant asymptote of a (rational) function is the line y = m + b such that as! ±,y! m + b. HA are special cases of SA. A horizontal asymptote is a slant asymptote, y = m + b, for which the slope is zero (and b is renamed k). Case III/ Parabolic Asymptote: y = Q The order of n factors). has the form y = a 2 + b + c ; a,b, c!r. is TWO GREATER THAN the order of d (with no common A parabolic asymptote of a (rational) function is the parabola y = a 2 + b + c such that as! ±,y! a 2 + b + c. HA and SA are special cases of PA. (What values of a and b from y = a 2 + b + c yield a HA? A SA?) Special Note: The difference between the orders of n and d can be any value. For eample, we can predict that y = 4 has (as its non-vertical asymptote) a + 3 CUBIC asymptote. In this course, we will restrict the types of non-vertical asymptotes to Horizontal and Slant (with perhaps an occasional BONUS involving Parabolic Asymptotes). Eercise: Determine the type and equation of the non-va of: a) y = 2 2 +! 6 b) y = 2 2 2 +! 6 c) y = 2 2! 6 Raelene Dufresne 2004 Rational Functions 6 of 4

Eercises: For each of the following, determine (where possible) a) the coordinates of any holes in the graph b) the coordinates of the and y intercepts c) the equation(s) of the vertical asymptote(s) d) the type and equation of the non-vertical asymptote. y = 2. y =! 3 3. y =! 4! 4 4. y = + 2 2! 4 5. y =! 2 + 3 6. y = 5! 2 2 + 2 7. y = 2! 3! 4 + 3 8. y = 2! 3! 4 Graphing Rational Functions 9. y =! 2 0. y = 2 6. y = 4! 2 2 2. y = 2 + 2!! 2 The two methods for graphing rational functions are by transformations and by elements. Before we discuss them, however, let s recall how to use the method of the Product of Signs to shade regions of the y plane (above and below the ais) where the graph WILL NOT LIE. Product of Signs Eercise: Perform the Product of Signs (Sign Chart) on (i) the quadratic function y = + 2 (!) and (ii) the rational function y = + 2!. a) Shade the regions where the graph does not lie. b) Are the shaded regions of the two functions the same or different? Eplain. c) Are the boundary regions of the two functions the same or different? Eplain. Raelene Dufresne 2004 Rational Functions 7 of 4

Transformational Method of Graphing Rational Functions To determine the transformations of any function, the functional equation must first be rewritten in transformational form, ( t y! k ) = f ( s! h % $ )', where the transformed # $ &' graph has To put rational functions into a horizontal reflection if s < 0. transformational form, we must recall the a vertical reflection if t < 0. Division Algorithm, which states that a horizontal stretch of s (if s! ). n a vertical stretch of t (if t! ). y = a horizontal translation of +h (if h! 0 ). d( )! y = Q R ( + d a vertical translation of +k (if k! 0). (these two forms are equivalent). Eample: Rewrite y = 5 + 2 + apply them to the basic rational function y = 5 Solution: +) 5 + 2, R =!3! y = 5 + 2 +! y = 5 3 + ( y! 5) =!3! ( 3 y! 5 ) = ( + ) ( + ) in transformational form. Then state the transformations, y = and sketch y = 5 + 2 +. y = 5 + #3 + y = 5 # 3 + VR VS of 3 HR: none VR: yes HS: none VS: of 3 HT: of VT: of 5! y = 5 + 2 + (after HT of & VT of 5) Raelene Dufresne 2004 Rational Functions 8 of 4

Special Note: There are different basic rational functions depending on the order of the denominator polynomial. So far, we have looked only at the basic rational function y = (the reciprocal). Other basic rational functions and their graphs include: Elemental Method of Graphing Rational Functions (and the Product of Signs!) Determine all of the elements of a rational function s graph by performing these steps in the following order.. Fully factor the numerator and denominator. If there are common factors, simplify the rational function and state the coordinates of the hole(s). Always work from the simplified rational function: 2. Determine the and y intercepts. 3. Determine the equation(s) of the vertical asymptote(s): set d = 0 and solve for. 4. Determine the (one!) equation and type of the non-vertical asymptote: determine the quotient of y = 2 =!2 n d and set y = Q. 5. Draw and y plane. Perform the Product of Signs on the (simplified) rational function and shade the regions where the graph WILL NOT LIE. 6. Plot all holes and intercepts. Sketch all asymptotes as dotted lines (the VA were already sketched when you did the boundary lines for the Product of Signs). Sketch the graph of the (simplified) rational function based on all of the elements that you have incorporated in the y plane. Eample: Sketch y = 5 + 2 + y = 3 =!3 y = 4 =!4 on graph paper using the elemental method outlined above. Raelene Dufresne 2004 Rational Functions 9 of 4

Eercises: Sketch each of the following using the elemental method. ALSO sketch #2, 3 and 5 by the transformational method. For every graph a) abel any relevant points/holes b) draw all asymptotes as dotted lines with arrows on the ends c) state the domain and range d) determine the symmetry of each function. y = 2. y =! 3 3. y =! 4! 4 4. y = + 2 2! 4 5. y =! 2 + 3 6. y = 5! 2 2 + 2 7. y = 2! 3! 4 + 3 8. y = 2! 3! 4 9. y =! 2 0. y = 2 6. y = 4! 2 2 2. y = 2 + 2!! 2 Elements and Graphs of Rational Functions: SOLUTIONS Equation, Special Points & Asymptotes Domain, Range & Symmetry Graph of Rational Function. y = int: none y int: none VA: = 0 HA y = 0 2. y =! 3 int: none y int: 0,! % $ ' # 3& VA: = 3 Domain:!R, 0 Range: y!r, y 0 ODD Domain:!R, 3 Range: y!r, y 0 Raelene Dufresne 2004 Rational Functions 0 of 4

3. y =! 4! 4 int: none y int: 0, VA: = 4 4. y = + 2 2! 4! y =! 2,! 2 Domain:!R, 4 Range: y!r, y 0 holes:!2,! % $ ' # 4 & int: none y int: 0,! % $ ' # 2& VA: = 2 Domain:!R, ±2 Range: y!r, y # 4, 0 5. y =! 2 + int: 2,0 y int: 0,!2 VA: =! HA: y = 3 6. y = 5! 2 int: none! y int: 0, 3 $ # & 5% VA: = 5 2 7. y = Domain:!R, # Range: y!r, y Domain:!R, 5 2 Range: y!r, y 0 2 + 2 2! 3! 4! y = 2! 4,! Raelene Dufresne 2004 Rational Functions of 4

holes:!,! 2 % $ ' # 5& int: none y int: 0,! % $ ' # 2& VA: = 4 + 3 8. y = 2! 3! 4 int:!3, 0 y int: 0,! 3 % $ ' # 4 & VA: =!, = 4 Domain:!R, #,4 Range: y!r, y # 2, 0 Domain:!R, #,4 Range: y!r 9. y =! 2! y =! + int:!, 0 y int: none VA: = 0 SA: y =!, (, 0) Domain:!R, 0 Range: y!r ODD 0. y = 2 int: none y int: none VA: = 0 Domain:!R, 0 Range: y!r, y 0 EVEN Raelene Dufresne 2004 Rational Functions 2 of 4

6. y = 4! 2 2 int: none! y int: 0, 3 $ # & 2% VA: =! 2, = 2 2 + 2. y = 2!! 2 int: none y int: 0,! % $ ' # 2& VA: =!, = 2 HA: y = Domain:!R, ± 2 Range: y!r, y 0 EVEN Domain:!R, #,2 Range: y!r, y 2 + Special Note: How do you know that the graph of y = in #2 will cross the HA 2!! 2 6 @ y =? How do you know that the graph of y = in # WON T cross the HA @ 2 4! 2 y = 0? For #, we know that there are no intercepts; therefore, setting y = 0 yields no solution (no intercepts). Therefore, there are no finite values in the graph of 6 y = 4! 2 for which the y value equals 0. Therefore, the graph of y = 6 in # 2 2 4! 2 WON T cross the HA @ y = 0. Checking to see if the graph crosses an HA @ y = 0 is automatic because we always check for intercepts (for which y = 0). We do not automatically solve for values that correspond to any other non-vertical asymptotes (for NON-ZERO values of y). For eample, we haven t performed such a check in #2 (or any of the graphs that do not have a horizontal asymptote @ y = 0, for that matter). To check if a graph crosses Raelene Dufresne 2004 Rational Functions 3 of 4

!# ANY non-vertical asymptote (horizontal, slant or otherwise), solve the system y = r $ # y = Q( ) for. If there is a solution, then the graph WILL cross the non-va at that coordinate. $ 2 + For #2, solve # y = 2 for. If there IS a solution for, the rational function!! 2 % $ y = 2 + y = and the horizontal asymptote y = intersect (the graph crosses the HA). 2!! 2 Substituting y = into y = 2 + 2!! 2 yields: = 2 + 2!! 2. Multiplying by the LCD yields: 2!! 2 = 2 +. Collecting like terms yields:!! 2 = +! 2! = =!3. Because the point!3, lies on the rational function (as WELL as on the HA), the graph of the rational function INDEED crosses the horizontal asymptote Raelene Dufresne 2004 Rational Functions 4 of 4