Sect Graphing Techniques: Transformations

Transcription

1 Sect. - Graphing Techniques: Transformations Recall the general shapes of each of the following basic functions and their properties: Identity Function Square Function f(x) = x f(x) = x Cube Function Square Root Function f(x) = x f(x) = x

2 Cube Root Function f(x) = x Reciprocal Function f(x) = x Absolute Value Function f(x) = x Greatest Integer Function f(x) = int(x)

3 Objective #: Graphing functions Using Vertical and Horizontal Shifts. Vertical Shift The graph g(x) = f(x) + k is the graph of f(x) shifted vertically by k units. Ex. h(x) = x Ex. g(x) = x + Since k =, we will Since k =, we will shift the graph of x down shift the graph of x by units. up by units Horizontal Shift The graph g(x) = f(x h) is the graph of f(x) shifted horizontally by h units. It is important to pay attention to the sign of h since the form is f(x h). Thus, the graph of g(x) = f(x ) will be shifted to the right by three units while the graph of r(x) = f(x + ) will be shifted to the left by two units since f(x + ) = f(x ( )). Ex. h(x) = (x ) Ex. g(x) = x + up

4 Since h =, we will Since h =, we will shift the graph of x to the shift the graph of x to the right by units. left by units Ex. g(x) = x + Ex. q(x) = (x +.7) This graph is the graph of x This graph is the graph of x shifted up and to the right. shifted down and to the left

5 Objective #: Graphing Functions Using vertical Compressions & Stretches. Vertical Stretch/Compression (shape) The graph of g(x) = a f(x) is the graph of f(x) stretched vertically by a factor of a if a > and compressed vertically by a factor of a if a <. Ex. 7 p(x) = x Ex. 8 r(x) = x The graph of p is the graph of x stretched by a factor of. a factor of. The graph of r is the graph of x compressed by Ex. 9 g(x) = x+ The graph of g is the graph of x stretched by a factor of and shifted down units and to the left two units. Ex. h(x) = int(x ) The graph of h is the graph of int(x) compressed by a factor of unit. and shifted to the right one

6 Objective #: Graphing Functions Using Reflections. Reflection The graph of g(x) = f(x) is the graph of f(x) reflected across the x-axis. The graph of h(x) = f( x) is the graph of f(x) reflected across the y-axis. Ex. g(x) = x Ex. h(x) = x The graph of g is the graph of x reflected across the x-axis. The graph of h is the graph of x reflected across the y-axis

7 General Strategy for transformations: In graphing g(x) = a f(x h) + k, we will start with the graph of f(x). In graphing g(x) = = a f( x h) + k, we will need to factor out from x h first and then start with the graph of f(x). We will then: i) Stretch it by a factor of a if a > ii) iii) or compress it by a factor of a if a <. Reflect it across the x-axis if a is negative. Reflect it across the y-axis if there is a minus sign in front of x. Shift it horizontally by h units and vertically by k units. Ex. r(x) =. (x+) + Ex. w(x) = x+ i) r is compressed by First, factor out : a factor of.. w(x) = (x ) ii) It is reflected across i) w is stretched by the x-axis. a factor of. iii) It is shifted left and ii) It is reflected across up. the y-axis. iii) It is shifted right & down. 7

8 7 Given the graph below, write the function: Ex. Ex The parent function is x. The parent function is x. It has been stretched by a It has been compressed by factor of and reflected across the x-axis, so a =. It has been shifted right and up. Thus, h = and k =. The function will be in the form of g(x) = a x h a factor of, but there is no reflection, so a =. It as been shifted left and down. Thus, h = and k =. The function will be in the form of + k. h(x) = a x h + k. So, the So, the function is function is h(x) = x ( ) g(x) = Objective #: x or h(x) = x + Graphing Functions Using horizontal Compressions & Stretches. Horizontal Stretch/Compression (shape) The graph of g(x) = f(ax) is the graph of f(x) compressed horizontally by a factor of /a if a > and stretched by a factor of /a if a <. For most of the basic functions we have worked with so far, there has not been a need for this since if we had a function like g(x) = (x), we are able

9 to move the factors of outside to write 9x and treat it a vertical stretch. In chapter, we will encounter functions that we cannot pull off this trick. Given the graph of f(x) below, find the following: Ex. 7 f(x) 8 a) f(x) b) f( x) a) The graph is compressed by a factor of : b) The graph is stretched by a factor of :