1. Transformation of Functions
Recognize graphs of common functions
Use translations to graph functions
Use reflections to graph functions
Use dilations to graph functions
Graph functions with sequence of transformations

2. Graphs of hyperbolas, square root function, polynomials.
The following basic graphs will be used extensively. It is important to be able to sketch these from memory.

3. The identity function f(x) = x

4. The quadratic function

5. The square root function

6. The absolute value function

7. The cube root function

8. We will now see how certain transformations of a function change its graph. This will give us a better idea of how to quickly sketch the graph of certain functions. The transformations are (1) translations, (2) reflections, and (3) dilations.

9. Vertical Translation Vertical Translation For b > 0,
the graph of y = f(x) + b is the graph of y = f(x) shifted up b units;
the graph of y = f(x)  b is the graph of y = f(x) shifted down b units.

10. Horizontal Translation
For d > 0,
the graph of y = f(x  d) is the graph of y = f(x) shifted d units right;
the graph of y = f(x + d) is the graph of y = f(x) shifted d units left.

11. Vertical shifts Horizontal shifts Moves the graph up or down.
Impacts only the “y” values of the function.
No changes are made to the “x” values.
Horizontal shifts
Moves the graph left or right.
Impacts only the “x” values of the function.
No changes are made to the “y” values.

12. Recognizing the translation from the equation, examples of shifting the function f(x) =
Vertical shift of 3 units up
Horizontal shift of 3 units left (HINT: x’s go the opposite direction that you might believe.)

13. Points represented by (x , y) on the graph of f(x) become
If the point (6, -3) is on the graph of f(x),
find the corresponding point on the
graph of f(x+3) + 2

14. Use the basic graph to sketch the following:

15. Combining a vertical & horizontal translation
Example of function that is shifted down 4 units and right 6 units from the original function.

16. Reflections
The graph of f(x) is the reflection of the graph of f(x) across the x-axis.
The graph of f(x) is the reflection of the graph of f(x) across the y-axis.
If a point (x, y) is on the graph of f(x), then (x, y) is on the graph of f(x), and
(x, y) is on the graph of f(x).

17. Reflecting Across x-axis (y becomes negative, -f(x).
Across y-axis (x becomes negative,
f(-x)). Even functions remain the same.

18. Use the basic graph to sketch the following:

19. Dilations The graph of af(x) can be obtained from the graph of f(x) by
stretching vertically for |a| > 1, or
shrinking vertically for 0 < |a| < 1.
For a < 0, the graph is also reflected across the x-axis.
(The y-coordinates of the graph of y = af(x) can be obtained by multiplying the y-coordinates of y = f(x) by a.)

20. DILATION FROM THE X-AXIS
y’s do what we think they should: If you see 3(f(x)), all y’s are MULTIPLIED by 3 (it’s now 3 times as high)

21. Dilation from the y-axis
The graph of y = f(cx) can be obtained from the graph of y = f(x) by
shrinking horizontally for |c| > 1, or
stretching horizontally for 0 < |c| < 1.
For c < 0, the graph is also reflected across the y-axis.
(The x-coordinates of the graph of y = f(cx) can be obtained by dividing the x-coordinates of the graph of y = f(x) by c.)

22. Dilation from the y-axis
We’re MULTIPLYING by an integer (not 1 or 0).
x’s do the opposite of what we think they should. (If you see 3x in the equation where it used to be an x, you DIVIDE all x’s by 3, thus it’s compressed horizontally.)

23. Sequence of transformations
Follow order of operations.
Select two points (or more) from the original function and move that point one step at a time.
1st transformation would be (x+2), which moves the function left 2 units (subtract 2 from each x),
pts. are now (-3,-1), (-2,0), (-1,1)
2nd transformation f(x) contains (-1,-1), (0,0), (1,1)
would be 3 times all the y’s,
pts. are now (-3,-3), (-2,0), (-1,3)
3rd transformation would be subtract 1 from all y’s, pts. are now (-3,-4), (-2,-1), (-1,2)

24. Graph of Example
(-1,-1), (0,0), (1,1)
(-3,-4), (-2,-1), (-1,2)

25. The point (-12, 4) is on the graph of y = f(x)
The point (-12, 4) is on the graph of y = f(x). Find a point on the graph of y = g(x).
g(x) = f(x-2)
g(x)= 4f(x)
g(x) = f(½x)
g(x) = -f(x)
(-10, 4)
(-12, 16)
(-24, 4)
(-12, -4)

26. Function notation to denote transformations

27. Challenge question
Point (2, 3) is on the graph of f(x).
Find the coordinates of this point when the following transformations have been applied:
a) 3f(x-1) c) –f(2x)
b) -2f(-x+4)+1