a translation by c units a translation by c units

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1 1.6 Graphical Transformations Introducing... Translations 1.) Set your viewing window to [-5,5] by [-5,15]. 2.) Graph the following functions: y 1 = x 2 y 2 = x y 3 = x y 4 = x 2-2 y 5 = x ) What effect do the +3, +1, -2 and -4 have? [Hmmm...perhaps these are the wrong kind of translations?] 4.) Graph the following functions: y 1 = x 2 y 2 = (x + 3) 2 y 3 = (x + 1) 2 y 4 = (x - 2) 2 y 5 = (x - 4) 2 5.) What effect do the +3, +1, -2 and -4 have? Aug 5 4:29 PM Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y = f(x): Vertical Translations y = f(x) + c y = f(x) - c a translation by c units a translation by c units Horizontal Translations y = f(x - c) y = f(x + c) a translation to the by c units a translation to the by c units Examples Describe how the graph of y = x can be transformed to the graph of the given equation. 1.) y = x ) y = x ) y = x + 4 Aug 5 4:40 PM 1

2 Examples Sketch the graphs of the following functions. 1.) f(x) = x ) g(x) = (x + 5) 3 3.) h(x) = x ) m(x) = x - 2 Aug 5 5:40 PM Now Introducing... Reflections 1.) Graph the following functions: y 1 = - x y 2 = -x What effect do the negative signs have? 2.) Graph the following functions: y 1 = -x 2 y 2 = (-x) 2 What effect do the negative signs have? Reflections The following transformations result in reflections of the graph of y = f(x). y = -f(x) : y = f(-x) : Aug 5 6:04 PM 2

3 1.) Find the equation of the reflection of f(x) = x 3-2x 2 + 5x - 9 over the x-axis and y-axis. Examples Describe how the graph of y = x 2 can be transformed to the graph of the given equation. Don't forget to think about the order of operations! 1.) y = -(x + 4) 2 2.) y = (50 - x) 2 3.) y = (x - 9) ) y = -(x + 1.4) 2-6 Aug 5 6:16 PM And Finally...Introducing Stretches and Shrinks 1.) Graph the following functions: y 1 = 2x 2 y 2 = (1/2)x 2 What effect do the 2 and 1/2 have? 2.) Graph the following functions? y 1 = (2x) 2 y 2 = [(1/2)x] 2 What effect do the 2 and 1/2 have? Aug 6 11:21 AM 3

4 Stretches and Shrinks Let c be a positive real number. Then the following transformations result in stretches or shrinks of the graph of y = f(x): Horizontal Stretches or Shrinks y = f x a stretch by a factor of c if c > 1 c a shrink by a factor of c if 0 < c < 1 Vertical Stretches or Shrinks y = cf(x) a stretch by a factor of c if c > 1 a shrink by a factor of c if 0 < c < 1 Aug 6 11:37 AM Examples Let f(x) = x 3-16x. Transform f(x) by: a.) a vertical stretch by a factor of 3. b.) a horizontal shrink by a factor of 1/2. Examples Describe how the graph of y = x can be transformed to the graph of the given equation. 1.) y = 2x 2.) y = 2 x 3.) y =.2x 4.) y =.2 x Aug 6 12:01 PM 4

5 Examples 1.) Describe how the graph of f(x) = (x + 2) 4 can be transformed into the graph of g(x) = -(x - 2) 4. 2.) Describe a basic graph and a sequence of transformations that can be used to produce a graph of y = -3(x - 2) ) Describe a basic graph and a sequence of transformations that can be used to produce a graph of y =.0625x - 5. Aug 6 12:15 PM Examples A graph G is obtained from the graph of y by the sequence of transformations indicated. Write an equation whose graph is G. 1.) y = x 3 : a shift to the left 3 units, a vertical stretch by a factor of 4 Would the graph of G change if we reversed the order of those transformations? 2.) y = x : a shift right 6 units, then a horizontal shrink by a factor of 1/3, and finally a shift up 7 units. Aug 6 12:21 PM 5

6 Examples Graph by hand and check with the graphing calculator. 1.) f(x) = -4 x ) h(x) = 2x Aug 6 1:03 PM Examples The graph of f is shown below. 1.) Sketch y = f(x + 1) ) Sketch y = 2f(x - 1) + 3. Aug 6 2:13 PM 6

7 3.) Sketch y = f(x). 4.) Sketch y = f( x ). Jul 10 7:53 AM Jul 11 9:25 AM 7

8 1.6 Homework Name: Describe how the graph of y = x 2 can be transformed to the graph of the given equation. 1.) 2.) 3.) 4.) Describe how to tranform the graph of f into the graph of g. 5.) 6.) Jul 10 8:53 PM Without your calculator, using your knowledge of the basic functions and transformations, graph the following. 7.) 8.) 9.) 10.) Jul 10 8:58 PM 8

9 11.) Find the equation of the reflection of f(x) across the x axis and y axis. f(x) = x 4 3x 3 + 2x ) Explain algebraically why the graph of an odd function is the same when reflected across the x axis as it is when reflected across the y axis. 13.) Transform f(x) = x 2 + x + 2 by... a.) a vertical shrink by 1/2 b.) a horizontal shrink by 1/2 Jul 10 9:02 PM The graph of f(x) is given. Graph the following transformations. f(x) 14.) 15.) 16.) Jul 10 9:07 PM 9