Chapter Transformations of Functions TOPICS.5.. Shifting, reflecting, and stretching graphs Smmetr of functions and equations TOPIC Horizontal Shifting/ Translation Horizontal Shifting/ Translation Shifting, Reflecting, and Stretching Graphs Much of the material in this section was introduced in Section., in our discussion of quadratic functions. You ma want to review the was in which the basic quadratic function f = can be shifted, stretched, and reflected as ou work through the more general ideas here. THEOREM Let f () be a function, and let h be a fied real number. If we replace with h, we obtain a new function g() = f ( h). The graph of g has the same shape as the graph of f, but shifted to the right b h units if h > and shifted to the left b h units if h <. EXAMPLE Sketch the graphs of the following functions. a. f = ( + ) b. g= Note: Solutions: Begin b identifing the underling a. function that is being shifted. The basic function being shifted is. Begin b drawing the basic cubic shape f (the shape of = ). Since is replaced b +, the graph of f b units. Note, for eample, that, on the graph. is the graph of shifted to the left is one point
Transformations of Functions Section.5 Vertical Shifting/ Translation Vertical Shifting/ Translation b. CAUTION! g The minus sign in the epression h is critical. When ou see an epression in the form + h ou must think of it as ( h). Consider a specific eample: replacing with 5 shifts the graph 5 units to the right, since 5 is positive. Replacing with + 5 shifts the graph 5 units to the left, since we have actuall replaced with 5. THEOREM Let f () be a function whose graph is known, and let k be a fied real number. The graph of the function g= f + k is the same shape as the graph of f, but shifted upward if k > and downward if k <. EXAMPLE Sketch the graphs of the following functions. a. f = + b. g= The basic function being shifted is. Start b graphing the basic absolute value function. The graph of g= has the same shape, but shifted to the right b units. Note, for eample, that (, ) lies on the graph of g.
Chapter Note: As before, begin b identifing the basic function being shifted. Horizontal and Vertical Shifting Note: In this case, it doesn t matter which shift we appl first. However, when functions get more complicated, it is usuall best to appl horizontal shifts before vertical shifts. Solutions: a. b. f EXAMPLE g Sketch the graph of the function f = + +. Solution: f The basic function being shifted is. The graph of f= + is the graph of = shifted up units. Note that this doesn t change the domain. However, the range is affected; the range of f is,,. The basic function being shifted is. Begin b graphing the basic cube root shape. To graph g=, we shift the graph of = down b units. The basic function being shifted is. Begin b graphing the basic square root shape. In f we have replaced with +, so shift the basic function units left. Then shift the resulting function unit up.
Transformations of Functions Section.5 Reflecting with Respect to the Aes Reflecting with Respect to the Aes Note: We state that a function is reflected with respect to particular ais. Visuall, this means the function is reflected over (across) that ais. THEOREM Given a function f :. The graph of the function g respect to the -ais. =. The graph of the function g= f respect to the -ais. f is the reflection of the graph of f with is the reflection of the graph of f with In other words, a function is reflected with respect to the -ais b multipling the entire function b, and reflected with respect to the -ais b replacing with. EXAMPLE Sketch the graphs of the following functions. a. f = b. g= Solutions: a. b. g f To graph f=, begin with the graph of the basic parabola =. The entire function is multiplied b, so reflect the graph over the -ais, resulting in the original shape turned upside down. Note that the domain is still the entire real line, but the range of f is the interval,. ( ] To graph g=, begin b graphing =, the basic square root. In g, has been replaced b, so reflect the graph with respect to the -ais. Note that this changes the domain but not the range. The domain of g is the interval, and the range is,. ( ] [ )
Chapter Vertical Stretching and Compressing Vertical Stretching and Compressing Note: When graphing stretched or compressed functions, it ma help to plot a few points of the new function. THEOREM Let f () be a function and let a be a positive real number.. The graph of the function g= af is stretched verticall compared to the graph of f if a >.. The graph of the function g af the graph of f if < a <. EXAMPLE 5 Sketch the graphs of the following functions. a. f = b. g = 5 Solutions: a. b. f 8 8 g = is compressed verticall compared to Begin with the graph of. The shape of f is similar to the shape of but all of the -coordinates have been multiplied b the factor of, and are consequentl much smaller. Begin with the graph of the absolute value function. In contrast to the last eample, the graph of g= 5 is stretched compared to the standard absolute value function. Ever second coordinate is multiplied b a factor of 5.
Transformations of Functions Section.5 5 Order of Transformations If the function g is obtained from the function f b multipling f b a negative real number, think of the number as the product of and a positive real number (namel, its absolute value). This is a simple eample of a function going under multiple transformations. When dealing with more complicated functions, undergoing numerous transformations, we need a procedure for untangling the individual transformations in order to find the correct graph. PROCEDURE If a function g has been obtained from a simpler function f through a number of transformations, g can be understood b looking for transformations in this order:. Horizontal shifts. Stretching and compressing. Reflections. Vertical shifts Consider, for eample, the function g= + +, which has been built up from the basic square root function through a variet of transformations.. First, has been transformed into + b replacing with +, and we know that this corresponds graphicall to a shift to the left of unit.. Net, the function + has been multiplied b to get the function +, and we know that this has the effect of stretching the graph of + verticall.. The function + has then been multiplied b, giving us +, and the graph of this is the reflection of + with respect to the -ais.. Finall, the constant has been added to +, shifting the entire graph upward b units. These transformations are illustrated, in order, in Figure, culminating in the graph of g= + +. + + + + + Figure : Building the Graph of g= + +
Chapter Order of Transformations EXAMPLE Sketch the graph of the function f =. Solution: The basic function that f is similar to is. Following the order of transformations:. If we replace b + (shifting the graph units to the left), we obtain the function, which is closer to what we want. +. There does not appear to be an stretching or compressing transformation.. If we replace b, we have + =, which is equal to f. This reflects the graph of with respect to the -ais. +. Since we have alread found f, we know there is no vertical shift. The entire sequence of transformations is shown below, ending with the graph of f. + Note: An alternate approach to graphing f = is to rewrite the function in the form f =. In this form, the graph of f is the graph of shifted two units to the right, and then reflected with respect to the -ais. The result is the same, as ou should verif. Rewriting an equation in a different form never changes its graph.
Transformations of Functions Section.5 7 TOPIC Smmetr of Functions and Equations -Ais Smmetr We know that replacing with reflects the graph of a function with respect to the -ais, but what if f ( )= f? In this case the original graph is the same as the reflection! This means the function f is smmetric with respect to the -ais. DEFINITION The graph of a function f has -ais smmetr, or is smmetric with respect to the -ais, if f f = for all in the domain of f. Such functions are called even functions. Figure : A Function with -Ais Smmetr Functions whose graphs have -ais smmetr are called even functions because polnomial functions with onl even eponents form one large class of functions 8 with this propert. Consider the function f = 7 5 +. This function is a polnomial of four terms, all of which have even degree. If we replace with and simplif the result, we obtain the function f again: 8 f ( )= 7( ) 5( ) + ( ) 8 = 7 5 + f = Be aware, however, that such polnomial functions are not the onl even functions. We will see man more eamples as we proceed. There is another class of functions for which replacing with results in the eact negative of the original function. That is, f ( )= f for all in the domain, and this means changing the sign of the -coordinate of a point on the graph also changes the sign of the -coordinate. What does this mean geometricall? Suppose f is such a function, and that f is a point on the graph of f. If we change the sign of both coordinates, we obtain a new point that is the original point reflected through the origin (we can also think of this as reflected over the -ais, then the -ais). (, )
8 Chapter Origin Smmetr For instance, if (, f ) lies in the first quadrant, (, f ) lies in the third, and if (, f ) lies in the second quadrant, (, f ) lies in the fourth. But since f ( )= f, the point (, f ) can be rewritten as (, f( ) ). Written in this form, we know that (, f( ) ). is a point on the graph of f, since an point of the form (?, f (?)) lies on the graph of f. So a function with the propert = has a graph that is smmetric with respect to the origin. f f DEFINITION The graph of a function f has origin smmetr, or is smmetric with respect to the origin, if f f = for all in the domain of f. Such functions are called odd functions. Figure : A Function with Origin Smmetr As ou might guess, such functions are called odd because polnomial functions with onl odd eponents serve as simple eamples. For instance, the function f = + 8 is odd: f ( )= ( ) + 8 = ( )+ 8 = 8 = f As far as functions are concerned, -ais and origin smmetr are the two principal tpes of smmetr. What about -ais smmetr? It is certainl possible to draw a graph that displas -ais smmetr; but unless the graph lies entirel on the -ais, such a graph cannot represent a function. Wh not? Draw a few graphs that are smmetric with respect to the -ais, then appl the Vertical Line Test to these graphs. In order to have -ais smmetr, if (, ) is a point on the graph, then (, ) must also be on the graph, and thus the graph can not represent a function. This brings us back to relations. Recall that an equation in and defines a relation between the two variables. There are three principal tpes of smmetr that equations can possess.
Transformations of Functions Section.5 9 Smmetr of Equations Smmetr of Equations Note: If ou don t know where to begin when sketching a graph, plotting points often helps ou understand the basic shape. DEFINITION We sa that an equation in and is smmetric with respect to:. The -ais if replacing with results in an equivalent equation. The -ais if replacing with results in an equivalent equation. The origin if replacing with and with results in an equivalent equation Knowing the smmetr of a function or an equation can serve as a useful aid in graphing. For instance, when graphing an even function it is onl necessar to graph the part to the right of the -ais, as the left half of the graph is the reflection of the right half with respect to the -ais. Similarl, if a function is odd, the left half of its graph is the reflection of the right half through the origin. EXAMPLE 7 Sketch the graphs of the following relations, making use of smmetr. a. f = Solutions: a. b. b. g= c. = This relation is a function, one that we alread graphed in Section.. Note that it is indeed an even function and ehibits -ais smmetr: f( ) = ( ) = = f While we do not et have the tools to graph general polnomial functions, we can obtain a good sketch of g=. (verif this). First, g is odd: g = g If we calculate a few values, such as g=, g = 8, g ()=, and g=, and then reflect these through the origin, we get a good idea of the shape of g.
7 Chapter c. Summar of Smmetr 8 The first column in the table below summarizes the behavior of a graph in the Cartesian plane if it possesses an of the three tpes of smmetr we covered. If the graph is of an equation in and, the algebraic method in the second column can be used to identif the smmetr. The third column gives the algebraic method used to identif the tpe of smmetr if the graph is that of a function f. Finall, the fourth column contains an eample of each tpe of smmetr. A graph is smmetric with respect to: The -ais if whenever the point (, ) is on the graph, the point (, ) is also on the graph. The -ais if whenever the point (, ) is on the graph, the point (, ) is also on the graph. The origin if whenever the point (, ) is on the graph, the point (, ) is also on the graph. If the graph is of an equation in and, the equation is smmetric with respect to: The -ais if replacing with results in an equivalent equation. The -ais if replacing with results in an equivalent equation. The origin if replacing with and with results in an equivalent equation. The equation = is not a function, but it is a relation in and that has -ais smmetr. If we replace with and simplif the result, we obtain the original equation: = ( ) = The upper half of the graph is the function =, so drawing this and its reflection gives us the complete graph of =. If the graph is of a function f (), the function is smmetric with respect to: The -ais if f f =. We sa the function is even. Not applicable (unless the graph consists onl of points on the -ais). The origin if f f =. We sa the function is odd. (, ) (, ) Eample: (, ) (, ) (, ) (, )
Transformations of Functions Section.5 7 Eercises For each function or graph below, determine the basic function that has been shifted, reflected, stretched, or compressed.. f = ( ) +. f = + 5. f = +. f = + 5. f= + 5. f = ( + ) + 7. f = + 8. f = ( + ) 9....
7 Chapter Sketch the graphs of the following functions b first identifing the more basic functions that have been shifted, reflected, stretched, or compressed. Then determine the domain and range of each function. See Eamples through.. f = ( + ). G= 5. p = ( + ) +. g= + 7. q = 9. s=. F=. v =. f = 5. b = + 5 8. g= + 8. r= + +. w= ( ). k= +. R= 7. S= 9. h =. W=. W=. S = +. V= +. g= + 9 (Hint: Find a better wa to write the function.) Write a formula for each of the functions described below. 5. Use the function g() =. Move the function units to the left and units down.. Use the function g() =. Move the function units to the right and units up. 7. Use the function g() =. Reflect the function across the -ais and move it units up. 8. Use the function g() =. Move the function units to the right and reflect across the -ais.
Transformations of Functions Section.5 7 9. Use the function g=. Move the function unit to the left and reflect across the -ais.. Use the function g=. Reflect the function across the -ais and move it down units.. Use the function g=. Move the function units to the right and units up.. Use the function g=. Move the function 5 units to the left and reflect across the -ais.. Use the function g=. Move the function 7 units to the left, reflect across the -ais, and reflect across the -ais.. Use the function g=. Move the function 8 units to the right, units up, and reflect across the -ais. Determine if each of the following relations is a function. If so, determine whether it is even, odd, or neither. Also determine if it has -ais smmetr, -ais smmetr, origin smmetr, or none of the above, and then sketch the graph of the relation. See Eample 7. 5. f = +. g= 7. h = 8. w= 9. = 5. = 5. + = 5. F= ( ) 5. = + 5. = 55. g= 5 5. m = 5 57. = 58. + =