2.3 TRANSFORMATIONS OF GRAPHS


 Gillian Jefferson
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1 78 Chapter Functions 7. Overtime Pa A carpenter earns $0 per hour when he works 0 hours or fewer per week, and timeandahalf for the number of hours he works above 0. Let denote the number of hours he works in a given week and (dollars) the corresponding pa. (a) Write a piecewise formula giving as a function of where (b) If his pa for the week is $1070, how man hours did he work? 7. Telephone Call Cost Suppose a telephone compan charges 0 cents for a call up to one minute, and 50 cents for each additional minute (or fraction thereof ). Let denote the number of minutes ou talk and (dollars) the corresponding cost. Then is a function of given b Int. (a) Checkseveralvaluesofto see that this formula gives what ou would epect. (b) Suppose ou do not want to spend more than $5.00 for a call. How long can ou talk? Useagraphindotmodewithadecimalwindow.. TRANSFORMATIONS OF GRAPHS Formal logic is an impoverished wa of describing human thought, and the practice of mathematics goes far beond a set of algorithmic rules... Mathematics ma indeed reflect the operations of the brain, but both brain and mind are far richer in their nature than is suggested b an structure of algorithms and logical operations. F. David Peat Relationships among graphs will be used throughout precalculus and calculus. Whole families of graphs can be related to each other through a few transforma tions. When we understand the properties of the graph of one particular function f, we can immediatel get information about domain and range, about intercepts and smmetr, for an function whose graph is the graph of f shifted up or down, right or left, reflected, squeezed or stretched. Sometimes we work with a famil related to one of the core graphs shown in the catalog in Figure, but more generall we simpl ask how the graphs of two functions are related to each other. [At thirteen] it was hard for me to imagine original mathematics, thinking of something that no one else had thought of before. When I went to college...i thought I might become a biologist. I was interested in man different things. I studied pscholog and philosoph, for instance. We didn t have grades, but we did have written evaluations. And I kept getting the message that m true talents didn t lie in subject X but in mathematics. William Thurston Vertical Shifts All of the transformations we consider can be justified algebraicall. For eample, the graph of a function f consists of all the points, whose coordinates satisf the equation. If, is on the graph of f, then the coordinates, 1 satisf the equation f 1. Each point, 1 is one unit below the point,, so we have an observation that applies to an graph. The graph of f 1 is obtained b shifting the graph of f down 1 unit. The same argument applies for an positive number c. Vertical shifts, c. 0 From the graph of f, thegraphof f cis shifted up c units, f c is shifted down c units. Although we can give an argument toeplain the effect of each transformation, we are more concerned with having ou do enough eamples to see for ourself what
2 . Transformations of Graphs 79 happens for each kind of transformation we eamine. Accordingl, we will show lots of graphs, but for our benefit, we strongl encourage ou to use our graphing calculator to draw each graph ourself. The first eample asks for graphs of vertical shifts of two core graphs. While it is good practice to graph such graphs on our calculator, ou should be able to draw these graphs without technolog. Look at the equation, recognize the graph as a vertical shift, and make a rough sketch. EXAMPLE 1 Vertical shifts Identif the function as a vertical shift of a core graph (Figure in Section.) and sketch. (a) 1,, 1 (b), 1, (a) Each graph is a vertical translation of the core parabola of Figure b. The first is shifted 1 unit up, the second units down, and the third is 1 up. The three graphs are labeled in Figure 9. (b) Each is a vertical shift of the core absolute value graph of Figure d. The absolute value graphs are shown in Figure 10. = +1 = = (0, ) (0, ) (a) Up 1 unit (b) Down units 1 (c) Up unit FIGURE 9 Vertical shifts of f = = +1 = (a) Down units (b) Up 1 unit FIGURE 10 Vertical shifts of f (c) Down unit
3 80 Chapter Functions Horizontal Shifts Some operations are applied to the outside of a function. For eample, f, f, f. The effect of such operations is to change the graph verticall. Other operations appl to the inside of the function, as f, f, f. In the equation f u, u is called the argument of the function. In contrast to operations that affect a graph verticall, we have the following useful observation. Outsideinside operations Operations applied to the outside of a function affect the vertical aspects of the graph. Operations applied to the inside (argument) of a function affect the horizontal aspects of the graph. EXAMPLE Horizontal shifts Sketch graphs of (a), 1, 1 (b) 1,, (a) The first two are obviousl horizontal shifts of the core parabola, units right and 1 unit left, respectivel. For the third function, we must recognize that 1 1,andsoshiftthegraphof right 1 unit. We have the three graphs labeled in Figure 11. We should note that the calculator will provide the same graph, whether written 1or 1, and we might recognize the graph as a shifted parabola onl after seeing the graph. (b) Be careful with parentheses; note the difference between Y X 1 (a vertical shift), and Y (X 1) (a horizontal shift). Each graph in this part is a horizontal shift of the core square root function. See Figure 1. =( ) (, 0) 1 (, 0) ) = + 1 ) = +1 = ( 1) (a) Right units 1 (b) Left unit (c) Right 1 unit FIGURE 11 Horizontal shifts of f
4 . Transformations of Graphs 81 = +1 = = + ( 1, 0) (, 0) = (a) Left 1 unit (b) Right unit FIGURE 1 Horizontal shifts of f (c) Left units (a) Core graph There are some important observations we must make in looking at the horizontal shifts in Eample. While the graph of shifts down from the graph of,thegraphof is shifted to the right, the opposite direction from what some people epect. It ma help to remember that the low point on the parabola occurs when 0, and on the parabola, 0 when. However ou choose to remember the relationships, we have the following. Horizontal shifts, c. 0 From the graph of f, thegraphof = f c is shifted left c units, f c is shifted right c units. Reflections (b) Vertical reflection Comparing the graphs of f and f,itisclearthatforanpoint, on the graph of f, the point, belongs to the graph of f.thatis,thegraphof f is obtained from the graph of f b tipping it upside down, or, in more mathematical terms, reflecting in the ais. Since multipling a function b 1 reflects the graph verticall, we would epect multiplication of the argument b 1 to reflect the graph horizontall, as the net eample shows. = EXAMPLE Horizontal and vertical reflections Sketch graphs of,,. (c) Horizontal reflection FIGURE 1 Reflections of f With a graphing calculator we see essentiall the graphs shown in Figure 1. The graph of is the top half of a parabola. More important at the moment are the relations with the other graphs. From the graph, thegraphof is a reflection in the ais,whilethegraphof is a reflection in the ais, as epected.
5 8 Chapter Functions Horizontal and vertical reflections From the graph of f, thegraphof f is a vertical reflection (in the ais), f is a horizontal reflection (in the ais). Dilations: Stretching and Compressing Graphs Multipling a function b a constant greater than 1 has the effect of stretching the graph verticall: if the point, belongstothegraphof f,thenthepoint,c is on the graph of cf. If the positive constant c issmallerthan1,then the number c is smaller than,sothegraphof cf is a vertical compression toward the ais. In a similar fashion, it can be seen that multipling the argument has the effect of compressing or stretching the graph horizontall, toward the ais. A stretching or compression is called a dilation of the graph. EXAMPLE Vertical dilations For the function f, describe how the graphs of f and 0. f are related to the graph of f. Usingagraphingcalculatorfor,wegetthegraphshowninFigure 1(a), with intercept points where, 0,. Tracing along the curve, we see that the left hump is just a little higher than, where 1., and the low point is located smmetricall through the origin (the graph is clearl the graph of an odd function). For the graphs of the other two, the shape is similar, and the intercept points are the same, but the graph of rises to a left hump well above, and the low point is below, twice as far awa from the ais as the graph of f.thegraphof 0. is squashed verticall toward the ais,andthehighandlowpointsweseearelessthan1unitawafromthe ais (a) =( ) (b) =( ) (c) = (0.)( ) FIGURE 1 Vertical dilations of f EXAMPLE 5 Horizontal dilations For the function f, sketch graphs of f, f, and f 0.5. The graph of f is the one from the previous eample, and is shown again in Figure 15(a). For the graph of f, we must replace each b,soweenter Y (X)^ (X), and similarl for f 0.5.
6 . Transformations of Graphs (a) = (b) =() () (c) = (0.5) (0.5) FIGURE 15 Horizontal dilations of f The graph of f has the same vertical rise and fall to the turning points, but the intercepts have been squeezed together; each is twice as close to the origin as for f.thegraphof f 0.5 is stretched horizontall. The intercept point that was at (, 0) has been moved outward to where 0.5 ; the intercept points are (, 0). = Dilations, c. 0 From the graph of f, thegraphof cf is a vertical stretch if c 1 (b a factor of c), vertical compression if c 1 (bafactorofc); f c is a horizontal compression if c 1, horizontal stretch if c 1. (1, ) =( 1) FIGURE 1 Translation of f = = (+) FIGURE 17 Translation and reflection of f Combining Transformations All the transformations we have considered can be combined, and if we are careful, we can predict the effect of several transformations on a graph of a function. In most instances, we take the operations from the inside out, looking first at anthing that affects the argument of the function. EXAMPLE Vertical and horizontal shifts Predict the effect on the graph of the function f in graphing (a) f 1 (b) f. Then check our prediction with a calculator graph. (a) f 1 1. From a parabola,thegraphof 1 isashift1unitright.thenfor 1, shift the graph downunits.theresultisaparabolawhoselowpointisat(1, ). A calculator graph shows the solid parabola in Figure 1. (b) The graph of is a parabola shifted units left. Then multipling b 1 reflects the graph in the ais, tipping it upside down. We have the solid parabola opening downward in Figure 17.
7 8 Chapter Functions EXAMPLE 7 Identifing transformed graphs The graph of a function f is given, together with three transformed graphs. Describe the transformations needed to get the given graph and write an equation for the function whose graph is shown. Check b graphing our function. (a) f abs (Figure 18) (b) f (Figure 19) (1, ) (1, 1) f() = (i) (, 0) (1, 1) (ii) (iii) FIGURE 18 Graphs for Eample 7a (a) The graph in Figure 18(i) is a vertical stretch b a factor of, since (1, 1) is sent to (1, ), so an equation for the transformed graph is.ifweuse adecimalwindow,wecantraceonthegraphof toseethat(1,)is on our graph, as desired. We note that in this instance, we could just as easil have obtained the transformed graph b compressing toward the ais, for which an equation would be. Since, the function can be described either wa. For the graph in Figure 18(ii), the absolute value graph is shifted 1 unit right (replace the argument b 1), and 1 unit down. An equation is 1 1, which we graph to check. In Figure 18(iii) the graph is tipped upside down (reflected in the ais) andshiftedup1unit.anequationis 1.
8 . Transformations of Graphs 85 (1, 1) f() = ( 1, 1) (0, 0) (i) ( 1, ) (ii) (iii) FIGURE 19 Graphs for Eample 7b (b) Forpanel(i)inFigure19,shiftthegraphof left 1 unit and down 1 unit, so an equation is 1 1.Wetraceonthegraphtoverifthe location of the given points. For the graph in panel (ii), reflect the graph of in the ais, (replace b ),andshiftup1unit.anequationis 1. For the third panel, reflect the graph of in the ais and shift up 1 unit, so an equation is 1. Verif b graphing this equation. Summar of Basic Transformations We list here the basic transformations we have introduced in this section. Basic transformations of the graph of f~!, c. 0 The transformations that affect a graph verticall are applied outside the function; transformations that change horizontal aspects are applied inside the function (to the argument). Vertical Horizontal f c, shift up f c, shift left f c, shift down f c, shift right f, reflect in ais f, reflect in ais cf, dilate verticall f c, dilate horizontall
9 8 Chapter Functions EXERCISES. Check Your Understanding Eercises 1 5 True or False. Give reasons. 1. If the function f has a positive zero and g f, thengmust have a positive zero.. If the function f has a zero between 1 and and g f, thengmust have a negative zero.. Ifthegraphsof f and g intersect in Quadrants I and III, then the graphs of f and g must intersect in Quadrants II and IV.. Ifthegraphsof f and g intersect in Quadrants II and IV, then the graphs of f and g must intersect in Quadrants I and III. 5. If the graph of f contains points in Quadrants III and IV, then the graph of f must also contain points in Quadrants III and IV. Eercises 10 Fill in the blank so that the resulting statement is true. If calculator graphs of f and g are drawn using 10, 10 10, 10, then the displa will show the graphs intersecting in Quadrant(s).. f 7, g f 5 7. f, g f 8. f, g f 9. f 5, g f f, g f Develop Master Eercises 1 Related Graphs The graph of a function f contains the points P, and Q, 5. Give the coordinates of two points on the graph of the function (a) g, (b) h. 1. g f 1 ; h f. g f ; h f. g f ; h f 0.5. g f ;h f 5. g f ; h f. g f ;h 1 f Eercises 7 1 Related Graphs For the function f (a) Determine a formula for g and simplif. (b) Draw calculator graphs of f and g on the same screen. (c) Write a brief statement describing how the graphs of f and g are related. 7. g f 8. g f 9. g f 10. g f 11. g f 1. g f Eercises 1 1 Related Line Graphs Thegraphoffis a line through points P and Q. Draw graphs of the given function. It is not necessar to find an equation for an of the functions before drawing a graph. (a) f (b) f (c) f. 1. P 1, and Q, 1 1. P, and Q 1, Eercises 15 1 Line Segment Graphs The graph of f is a line segment joining P and Q. Draw a graph and give the domain and range of (a) f (b) f (c) f. 15. P, and Q, 1. P, and Q 0, Eercises Line Segment Graphs The graph of f is two line segments PQ and QR. Draw a graph and give the domain and range of (a) f (b) f (c) f. 17. P, 1, Q,, andr, P,, Q 1,, andr, 1 Eercises 19 Verbal Description Give a verbal descriptionofhowouwoulddrawagraphofgfromthe graph of f. Check b drawing the graphs. 19. f 1, g 1 0. f, g 1 1. f, g 1. f, g 0.5 Eercises Verbal to Formula A verbal description of transformations of the graph of f is given, resulting in a graph of function g. Give a formula that describes the function g. Confirm b drawing graphs of f and g on the same screen.. f. Translatethegraphoffto the left units and then reflect about the ais.. f. Translate the graph of f to the left units and then reflect about the ais. 5. f 1. Stretch the graph of f verticall upward bafactorof,thentranslatedownwardunits.. f 1. Compress the graph of f verticall downward b a factor of 0.5, then reflect about the ais. Eercises 7 0 Graph to Verbal and Formula The graph of function f is shown along with graphs of transformed functions g and h. (a) Give a verbal description of the transformations that will give the graphs of g and h from the graph of f. (b) Give a formula for g. Do the same for h. (c) Asacheck,drawgraphsofourformulasinpart(b) and see if the agree with the given graphs.
10 . Transformations of Graphs f() = +1 (, 0) (, 1) = h() 9. (1, ) (, 1) = g() (1, 1) f() = = h() (0, ) (1, 1) (, 1) = g() (0, ) 8. (1, 1) f() = (, 0) (, 1) = h() 0. (, 0) (, 0) = g() f() =
11 88 Chapter Functions (0, ) 9. f ; g f 0. f ; g f 1. f ; g f. f ; g f. Eplore For each number k draw a calculator graph of k on the same screen. From what ou observe write a brief paragraph comparing the graphs for different values of k. (a) k 1 (b) k (c) k 0.5. Follow instructions of Eercise for k Use the graph of f shown to sketch the graph of f 1. Label the coordinates of four points that must be on our graph. = g() (, 0) ( 1, 0) (, 0) = h() (0, ) (, ) (, ) (0, ) Eercises 1 Domain and Range The domain D and range R of function f are given in interval notation. Give the domain and range of the function (a) g, (b) h. 1. D,, R, 8 ; g f 1, h f. D, 5, R, 5 ; g f, h f. D,,R, ; g f,h f. D,, R, 8 ; g f,h f Eercises 5 8 Related Domain and Range The domain D and range R of function g are given. Find the domain and range of function f. 5. g f ; D,, R, 5. g f 1 ; D,,R, 7. g f ; D,, R 1, 8. g f ; D, 8, R, Eercises 9 Related Intercept Points (a) Determine the coordinates of the intercept points of the graph of function f. (b) Find the intercept point(s) of the graph of function g. (c) What is the intercept point for g? (d) Draw graphs as a check. (, 0) ( 1, ) (1, ) = f(). Use the graph of f shown to sketch the graph of f 1. Label the coordinates of four points that must be on our graph. (, 0) = f() 7. Points on Related Graphs Points P, and Q., 5. are on the graph of f 1.Givethe coordinates of two points that must be on the graph of (a) f (b) f.
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