1. 1. Transformations
To graph: Identify parent function and adjust key points.
Function
To Graph:
Move key point (x,y) to:
Vertical Shift up
Vertical Shift down
Horizontal Shift left
Horizontal Shift right
Reflection about x-axis
Reflection about y-axis
Vertical stretch if
Vertical shrink if
Horizontal stretch if 0 < b <1
Horizontal shrink if b > 1

2. Warm-up. For each function below,
a) state the domain b) even/odd/neither c) symmetry

3. Warm-up. Suppose 1) If , what is x?
2) Find all intercepts of the graph of f

4. Warm-up.
Suppose and are points on a line.
Write the equation of the line containing these 2 points.

5. Warm-up. 1. Evaluate the following: State the domain for this function
Sketch the graph

6. 2.6 Function Transformations

7. 2.6 Function
Transformations

8. a. Vertical Shift Parent function : Shift Down 2 units
Vertical Shift (or translation)
shifts UP k units
shifts DOWN k units

9. b. Horizontal Shift Parent function : Shift left 3 units
Horizontal shift (or translation)
shifts LEFT h units
shifts RIGHT h units

10. 2a. Reflection about the x-axis
Parent function :
Reflect over x-axis.
Reflects graph about the x-axis

11. 2b. Reflects graph about the y-axis
Parent function :
Reflect over y-axis.
Reflects graph about the y-axis

12. 3a. Stretch (dilate) the graph vertically
Parent function :
Stretch vertically by : 2
If a > 1, stretches graph vertically
If 0 < a < 1, compresses graph vertically

13. 3b. Horizontal Stretch/Compress
Horizontal Scale
If b > 1, compresses horizontally (x-values by 1/b)
If 0 < b < 1, stretches horizontally (x-values by 1/b)

14. 3b. Horizontal Dilation (Scale)
When scale is “inside” the parent function,
it is preferable to pull it OUTSIDE the parent function and apply vertical dilation

15. Practice

16. 4. Sequence of Transformations
When a function has multiple transformatinos applied, does the order of the transformations matter?
Which operation is first: Reflection or Shift ?

17. 5. a) Rewrite function in standard form
Step 1: Always, factor out coefficients and write in standard form, before doing transformations!
Rewrite in standard form:

18. Perform the transformations in this order1.
Vertical scale
Vertical shift 4.
Horizontal scale 2.
Horizontal shift 3.

19. 6. Describe sequence of Transformations
Standard Form:
Parent Function
Reflection over x-axis
Reflection over y-axis
Scale y
Scale x
Shift L/R
Shift U/D

20. 6. Describe sequence of Transformations
Standard Form:
Parent Function
Reflection over x-axis
Reflection over y-axis
Scale y
Scale x
Shift L/R
Shift U/D

21. 6. Describe sequence of Transformations
Standard Form:
Parent Function
Reflection over x-axis
Reflection over y-axis
Scale y
Scale x
Shift L/R
Shift U/D

22. 6. More Practice…
For each function, describe (in order) the sequence of transformations and sketch the final graph.
1) )
2) ) 3)

23. 7. Domain
How is the domain of a function affected by the transformations?

24. 11. Write an equation from the graph
Identify parent (shape)
Compare key points to determine if y values are scaled.
Observe translations and reflections
Write in standard form

25. “Slope” = 1 Move: Right 1, Up 1 to next point on graph
1. Library of Functions (Take note of key points)
“Slope” = 1
Move:
Right 1, Up 1
to next point on graph

26. College Algebra Notes 2.6 Write the Function from the Graph
For each graph below:
a)Name the parent function b) Describe the sequence of transformations (in order)
c) Determine the function that describes the graph
d) Verify key points by plugging into your function.
1) )

27. 3) )

28. 11. Write an equation from the graph

29. Transformations 1) 2) 3)

30. Warm-up.
a) List the sequence of transformations and sketch
b) List the transformations that are made to each key point of the parent function.
Even or Odd ?

31. 8. A second method for sequence of transformations
Method 2: Less Preferred method
When a function is not in the standard form, perform transformations in this order:
Horizontal shift
Stretch/shrink
Reflect
Vertical stretch Shrink

32. Perform the transformations in this order1.
Vertical scale by a
If a is negative, reflects across x-axis
Vertical shift
+k: shift up k
-k : shift down k 4.
Horizontal scale by
If b is negative, reflects across y-axis 2.
Horizontal shift
-h : shift to right
+h : shift to left 3.