Transformation of Functions You should know the graph of the following basic functions: f(x) = x 2. f(x) = x 3

Transcription

1 Transformation of Functions You should know the graph of the following basic functions: f(x) = x 2 f(x) = x 3

2 f(x) = 1 x f(x) = x

3 f(x) = x If we know the graph of a basic function f(x), we can draw the graph of more complicated functions, like 2f( x+1), using the method of transformation of functions. In order to do that, we do algebra to change f(x), one step at a time, to the new function. Each algebraic change causes a corresponding geometrical change to the graph of the previous function. We discuss the different types of geometrical changes that would occur to a function when we perform certain type of algebraic operation to a function:

4 Vertical Shift: Given a function f(x), if c > 0 is a constant, then: The graph of f(x) + c is the graph of f moved up c units. The graph of f(x) c is the graph of f moved down c units. Example: Starting with f 1 (x) = x 2, then f 2 (x) = f 1 (x) + 3 = x The graph of f 2 (x) = x is the graph of x 2 moved up by 3 units. x x 2

5 Example: Starting with f 1 (x) = x 2, we can obtain f 2 (x) = f 1 (x) 1 = x 2 1. The graph of x 2 1 is the graph of x 2 moved down by 1 unit. x 2 x 2 1

6 Horizontal Shift Again assuming c > 0. The graph of f(x + c) is the graph of f moved left c units. Notice that the graph of f actually moves to the left if we add a positive constant to the argument (input) of f. The graph of f(x c) is the graph of f moved right c units. Notice that the graph of f actually moves to the right if we subtract a positive constant from the argument (input) of f. Example: Starting with f 1 (x) = x, we can obtain f 2 (x) = f 1 (x 3) = x 3. The graph of x 3 is the graph of x moved to the right by 3 units. Notice that the new graph is actually moved to the right when a negative constant is added to the argument of a function. x x 3 Example: Starting with f 1 (x) = x, we can obtain f 2 (x) = f 1 (x + 1) = x + 1. The graph of x + 1 is the graph of x moved to the left by 3 units. Notice that the new graph is actually moved to the left when a positive constant is added to the argument of a function. x + 1 x

7 Vertical stretch/compression The graph of cf(x) is the graph of f stretched vertically (from the x axis) by a factor of c if c > 1. To visualize a vertical stretch, imagine that you are pulling the graph of f away from the x axis in both up and down directions. In a vertical stretch, the x intercepts are unchanged. Example: Starting with f 1 (x) = x, we can obtain f 2 (x) = 2f 1 (x) = 2 x. The graph of 2 x is the graph of x stretched by a factor of 2 vertically. 2 x x

8 The graph of cf(x) is the graph of f compressed vertically by a factor of 1 c if c < 1 To visualize a vertical compression, imagine you push the graph of f toward the x axis from both up and down directions. In a vertical compression, the x intercepts are unchanged. Example: Starting with f 1 (x) = x, we can obtain f 2 (x) = 1 3 f 1(x) = 1 3 x. The graph of 1 3 x is the graph of 3 compressed by a factor of 3 vertically. x 1 x 3

9 Horizontal stretch/compression The graph of f(cx) is the graph of f compressed horizontally by a factor of c if c > 1. To visualize a horizontal compression, imagine that you push the graph of the function toward the y axis from both the left and the right hand side. In a horizontal compression, the y intercept is unchanged. Example: Starting with f 1 (x) = 1 x, we can obtain f 2(x) = f 1 (2x) = 1 2x. The 1 graph of 2x is the graph of 1 compressed by a factor of 2 horizontally. x 1 x 1 2x

10 The graph of f(cx) is the graph of f strectched horizontally by a factor of 1 c if c < 1. To visualize a horizontal stretch, imagine that you pull the graph of the function away from the y axis from both the left and the right hand side. In a horizontal stretch, the y intercept is unchanged. ( ) ( ) Example: Starting with f 1 (x) = x 3, we can obtain f 2 (x) = f 1 2 x = 2 x. ( ) 3 1 The graph of 2 x is the graph of x 3 stretched by a factor of 2 horizontally. x 3 ( ) x Again, pay attention to the fact that, in a horizontal stretch/compression that, if the constant c multiplied to x is greater than 1, the resulting graph is actually compressed by a factor of 1/c. On the other hand, if c is less than 1, the resulting graph is stretched by a factor of c.

11 Reflection The graph of f(x) is the graph of f reflected with respect to the x-axis. To draw a reflection with respect to the x axis, draw the resulting graph by reflecting everything in the original graph using the x axis as the mirror. Example: Starting with f 1 (x) = x 2, we can obtain f 2 (x) = f 1 (x) = x 2. The graph of x 2 is the graph of x 2 reflected with respect to the x axis: x 2 x 2 Example: Starting with f 1 (x) = x, we can obtain f 2 (x) = f 1 (x) = x. The graph of x is the graph of x reflected with respect to the x axis: x x

12 The graph of f( x) is the graph of f reflected with respect to the y-axis. To draw a reflection with respect to the y axis, draw the resulting graph by reflecting everything in the original graph using the y axis as the mirror. Example: Starting with f 1 (x) = x, we can obtain f 2 (x) = f 1 ( x) = x. The graph of x is the graph of x reflected with respect to the y axis: x x Example: Starting with f 1 (x) = x 3, we can obtain f 2 (x) = f 1 ( x) = ( x) 3. The graph of ( x) 3 is the graph of x 3 reflected with respect to the y axis: ( x) 3 x 3

13 To draw the graph of a complicated function, start with a simple function whose graph is known. Algebraically change the known function one step at the time, by adding or subtracting a constant, multiply by a constant...etc, to the function in question. The geometry must be dictated by the algebraic changes, not the other way around. You must first finish the algebra as to how the simple function changes to the complicated function, then do the geometry. Example: Graph f(x) = 2x Ans: We start with f 1 (x) = x. We know the graph of f 1 (x) Let f 2 (x) = f 1 (x + 1) = x + 1. The graph of f 2 (x) can be obtained from the graph of f 1 (x) by a horizontal shift one unit to the left. f 2 (x) = x + 1 f 1 (x) = x f 3 (x) = f 2 (2x) = 2x + 1. The graph of f 3 (x) can be obtained by horizontally compress f 2 (x) by a factor of 2. f 3 (x) = 2x + 1 f 2 (x) = x + 1

14 f 4 (x) = f 3 ( x) = 2( x) + 1 = 2x + 1. The graph of f 4 (x) can be obtained by reflecting the graph of f 3 (x) with respect to the y axis. f 4 (x) = 2x + 1 f 3 (x) = 2x + 1 f 5 (x) = f 4 (x) 1 = 2x The graph of f 5 (x) is obtained by vertically shift f 4 (x) one unit down. f 4 (x) = 2x + 1 f 5 (x) = 2x The graph of f is obtained from the graph of x by moving left one unit, followed by horizontal compression by a factored of 2, reflected with respect to the y axis, then moved down one unit.

15 In the previous example, it is possible to change the order of how you would change the functions, but you must let the algebra dictate the geometry. Second approach to the same problem: Let g 1 (x) = x g 2 (x) = g 1 (2x) = 2x. The graph of g 2 (x) can be obtained by horizontally compress g 1 (x) by a factor of 2. g 2 (x) = 2x g 1 (x) = x g 3 (x) = g 2 ( x) = 2( x) = 2x. The graph of g 3 (x) can be obtained by reflecting g 2 (x) with respect to the y axis. g 3 (x) = 2x g 2 (x) = 2x

16 ( g 4 (x) = g 3 x 1 ) 2 = 2 ( x 1 ) 2 = 2x + 1. The graph of g 4 (x) is obtained by moving g 3 (x) half unit to the right. g 4 (x) = 2x + 1 g 3 (x) = 2x Notice in this step that we added 1 2 to the argument of g 3, not 1. The reason is because if we added 1 to the argument of g 3, the distributive property would give us 2x 2, which is not what we want. You must always actually work out the algebra of how each subsequent function is obtained from the previous function. g 5 (x) = g 4 (x) 1 = 2x The graph of g 5 (x) is obtained by moving the graph of g 4 (x) 1 unit down. g 4 (x) = 2x + 1 g 5 (x) = 2x The graph of f can be obtained from the graph of x by first horizontally compress by a factor of 2, reflected with respect to y, moved to the right by 1 2 unit, then moved down 1 unit.

17 Example: Draw the graph of 2 3x 1 Ans: We start with f 1 (x) = 1 x f 2 (x) = f 1 (x 1) = 1 x 1. The graph of g 2(x) can be obtained by horizontally shift f 1 (x) 1 unit to the right. f 1 (x) = 1 x f 2 (x) = 1 x 1

18 1 f 3 (x) = f 2 (3x) = (3x) 1 = 1 3x 1. The graph of f 3(x) can be obtained by horizontally compress f 2 (x) by a factor of 3. f 3 (x) = 1 3x 1 f 2 (x) = 1 x 1

19 ( ) 1 f 4 (x) = 2f 3 (x) = 2 = 2 3x 1 3x 1. The graph of f 4(x) can be obtained by vertically stretch f 3 (x) by a factor of 2. f 4 (x) = 2 3x 1 f 3 (x) = 1 3x 1

20 f 5 (x) = f 4 (x) = 2 3x 1. The graph of f 4(x) can be obtained by reflecting f 4 (x) with respect to the x axis. f 5 (x) = 2 3x 1 f 4 (x) = 2 3x 1

21 The method of transformation of functions can be applied to any function in general. As long as we have the graph of f, we can draw the graphs of other functions that are transformed from f. We do not need to have an expression explicitely defined for f. Example: The graph of a function f is drawn below. Draw the graph of 2f( 2x + 1) + 1 f(x)

22 Ans: We change f(x) using algebra and in each stage keep track of the change we made and the corresponding geometric changes: f 1 (x) = f(x + 1). The graph of f 1 (x) is obtained by horizontally shift the graph of f(x) 1 unit to left. f 1 (x) f(x) f 2 (x) = f 1 (2x) = f(2x + 1). compress f 1 (x) by factor of 2. The graph of f 2 (x) is obtained by horizontally f 2 (x) f 1 (x)

23 f 3 (x) = f 2 ( x) = f(2( x) + 1) = f( 2x + 1). The graph of f 3 (x) is obtained by reflecting the graph of f 2 (x) with respect to the y axis. f 3 (x) f 2 (x) f 4 (x) = 2f 3 (x) = 2f( 2x + 1). stretch f 3 (x) by factor of 2 The graph of f 4 (x) is obtained by vertically f 4 (x) f 3 (x)

24 f 5 (x) = f 4 (x) = 2f( 2x + 1). The graph of f 5 (x) is obtained by reflecting the graph of f 4 (x) with respect to x axis. f 4 (x) f 5 (x) f 6 (x) = f 5 (x)+1 = 2f( 2x+1)+1. The graph of f 6 (x) is obtained by vertically shift f 5 (x) 1 unit up. f 6 (x) f 5 (x)

25 Below is a graph of the original function f and the final result, 2f( 2x + 1) + 1 2f( 2x + 1) + 1 f(x)