2.4 Transformations of Functions
Suppose the graph of f is given. Describe how the graph of each of
the following functions can be obtained from f.
5. (a) y = -f(x) + 5
(b) y = f(-x) + 5
(a) f reflects the x-axis and moves up 5 units
(b) f reflects the y-axis and moves up 5 units
7. (a) y = f(x - 2) – 3 (b) y = 2f(x - 3)
(a) f moves 2 units to the right and moves down 3 units
(b) f stretches vertically by a factor of 2 and moves 3 units
to the right
8. (a) y =
1
f(x) + 10
2
(b) y =
1
f(x + 10)
2
(a) f shrinks vertically by a factor of ½ and moves up 10 units
(b) f shrinks vertically by a factor of ½ and moves to the left 10 units
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The graph of f is given. Sketch the graph of each of the following
functions.
12. y = -f(x) + 3
14. y =
2
1
f(x - 1)
2
Sketch the graph of the function, not by plotting points, but by
starting with the graph of a standard function and applying
transformations.
15. f(x) = ( x − 2)2
17. f(x) = −( x + 1) 2
19. f(x) = x3 + 2
3
22. f(x) = 2 − x + 1
24. f(x) = x + 2 + 2
4
Determine whether the function is even, odd, or neither using both
symmetry AND what you know about the definitions of even and odd
functions.
26. f(x) = x −2 f(x) =
f(-x) =
1
1
= 2
2
( − x)
x
1
x2
Since f(-x) = f(x), f is an even function
(* Note symmetry on graph of f, symmetry to y-axis)
29. f(x) = 3 x 3 + 2 x 2 + 1
f(-x) = 3(-x)3 + 2(-x)2 + 1 = -3x3 + 2x2 + 1
-f(x) = -(3x3 + 2x2 + 1) = -3x3 - 2x2 - 1
f(-x) does not equal f(x) or –f(x), therefore f is neither even nor odd
30. f(x) = x +
1
x
1
1
= −x −
(− x)
x
1
1

-f(x) = −  x +  = − x −
x
x

f(-x) = (− x) +
Since f(-x) = -f(x), f is an odd function
(* Note symmetry on graph of f, symmetry to origin)
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