1. 2.4 Transformations of Functions

2. Fill out the chart as we go along…
Transformation
f(x)

3. Vertical Shift
y = f(x) + c or y = f(x) – c up ‘c’ units down ‘c’ units EX: y = x2 and y = x2 - 2
F(x) x y -2 4 -1 1 2
F(x)-2 x y -2 -1 1 2

4. Fill out the chart as we go along…
Transformation
f(x)
Vertical Shift
f(x)+c

5. Horizontal Shift EX g(x) = (x + 4)2 y = f(x + c) or y = f(x – c)
left ‘c’ units right ‘c’ units EX
g(x) = (x + 4)2
f (x) x y -2 4 -1 1 2
f (x+4) x y

6. Fill out the chart as we go along…
Transformation
f(x)
Vertical Shift
f(x)+c
Horizontal Shift
f(x-c)

7. Graph: y = (x – 2)2 + 3
f (x) x y -2 4 -1 1 2
f (x-2)+3 x y

8. Reflecting Graphs
y = f(x) or y = -f(x)
The y-coordinate of each point of the graph of y = -f(x) is the negative of the y- coordinate of the corresponding on y = f(x).
Reflection in the x-axis.
f (x) x y -2 4 -1 1 2
-f (x) x y -2 -1 1 2

9. Fill out the chart as we go along…
Transformation
f(x)
Vertical Shift
f(x)+c
Horizontal Shift
f(x-c)
Reflection across x-axis
-f(x)

10. y = f(-x) Reflection in the y-axis-4
DNE -1 1 4 2 x y

11. Fill out the chart as we go along…
Transformation
f(x)
Vertical Shift
f(x)+c
Horizontal Shift
f(x-c)
Reflection across x-axis
-f(x)
Reflection across y-axis
f(-x)

12. Vertical Stretch & Shrink y= cf(x) (y-coordinate multiplied by ‘c’)
If c > 1 – stretch by a factor of ‘c’
If 0 < c < 1 – shrink vertically by a factor of ‘c’

13. Fill out the chart as we go along…
Transformation
f(x)
Vertical Shift
f(x)+c
Horizontal Shift
f(x-c)
Reflection across x-axis
-f(x)
Reflection across y-axis
f(-x)
Vertical stretch
cf(x)
if c>1, stretch
if 0<c<1, shrink

14. EX x y -2 4 -1 1 2 x y x y -2 -1 1 2

15. EX x y -4
DNE -1 1 4 2 x y

16. Horizontal Stretch & Shrink y = f(cx) (x-coordinate multiplied by ‘c’)

17. Fill out the chart as we go along…
Transformation
f(x)
Vertical Shift
f(x)+c
Horizontal Shift
f(x-c)
Reflection across x-axis
-f(x)
Reflection across y-axis
f(-x)
Vertical stretch
cf(x)
if c>1, stretch
if 0<c<1, shrink
Horizontal Stretch
f(cx)
if c>1, shrink
if 0<c<1, stretch

18. Ex Given: find y = f(2x) and y = f(1/2x)

19. EX x y -2 4 -1 1 2 x y x y

20. Even and Odd Functions Even if f(-x) = f(x) Odd if f(-x) = -f(x)
Symmetric with respect to y-axis
Odd if f(-x) = -f(x)
Symmetric with respect to the origin
(rotate 180º about the origin or reflect 1st in x-axis and then in y-axis.)

21. Ex even/odd/neither
f(x) = x3 + x
f(x) = 7 – x6
f(x) = 3x – x3

22. Make sure your chart is complete!!
Transformation
f(x)
Vertical Shift
f(x)+c
Horizontal Shift
f(x-c)
Reflection across x-axis
-f(x)
Reflection across y-axis
f(-x)
Vertical stretch
cf(x)
if c>1, stretch
if 0<c<1, shrink
Horizontal Stretch
f(cx)
if c>1, shrink
if 0<c<1, stretch

23. Homework
Pg 191
#1-35 odd, 41, 43, 61-66