1. Transformations: Shifts

2. Warm Up Find y-intercept, domain, range of
Given two points (0, -2) and (-1, -8) of the exponential graph, write the equation.
Growth or Decay? b)
4. Describe the Shift: (Up, Down, Left, Right)

3. Standard
Standard: MCC9-12.F.BF.3 Identify the effects on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs.

4. Learning Target
· I can describe the effect on the graph of f(x) by using the transformation f(x●k), where k is positive or negative.
· I can describe the effect on the graph of f(x) by using the transformation k ● f(x), where k is positive or negative.

5. Check for Understanding
The graph of f(x) = 2x is shown to the right. Determine the coordinates of point Q after the transformation g(x) = f(x - 3). 
Given the graph and the original function f(x) = 2x, find h and the new function. Describe the translation. (The original graph is in green and the transformed graph is in red.)

6. Check for Understanding
Describe how the graph of y = 4x is translated to get the graph of y = 4(x − 1) – 2.
The function g(x) is obtained by translating f(x) = 2x + 3 up 5 units. Write an equation for g(x). Do NOT simplify.

7. Mini Lesson
One type of non-rigid transformation is a stretch or compression. A vertical stretching is the stretching of the graph away from the x-axis. A vertical compression is the squeezing of the graph towards the x-axis. A compression is a stretch by a factor less than 1.

8. Mini Lesson
Vertical Stretches and Compressions For the parent function y = f(x), the vertical stretching or compression of the function is a ● f(x). If | a | < 1 (a fraction between 0 and 1), then the graph is compressed vertically by a factor of a units. If | a | > 1, then the graph is stretched vertically by a factor of a units. *For values of a that are negative, then the vertical compression or vertical stretching of the graph is followed by a reflection across the x-axis. To determine the coordinates of a specific point under a vertical stretch or compression, the x-value remains the same and the y-value is multiplied by a.

9. Mini Lesson

10. Mini Lesson
A horizontal stretching is the stretching of the graph away from the y-axis. A horizontal compression is the squeezing of the graph towards the y-axis.

11. Mini Lesson Horizontal Stretches and Compressions
A compression is a stretch by a factor less than 1.
If | b | < 1 (a fraction between 0 and 1), then the graph is stretched horizontally by a factor of b units. If | b | > 1, then the graph is compressed horizontally by a factor of b units.
*For values of b that are negative, then the horizontal compression or horizontal stretching of the graph is followed by a reflection across the y- axis.
To determine the coordinates of a specific point under a horizontal stretch or compression, the y-value remains the same and the x-value is divided by b.

12. Mini Lesson

13. Mini Lesson
Multiple transformations may be performed on one parent graph, that is if y = f(x), then f(x – h) + k will translate the graph horizontally h units and vertically k units.

14. Work Session
Example 1. Point M is on the graph of f(x) = (1/2)x, as shown. Determine the coordinates of point M under the transformation 5/4f(x).

15. Work Session
Example 1. Point M is on the graph of f(x) = (1/2)x, as shown. Determine the coordinates of point M under the transformation 5/4f(x).
Solution: 
g(x) = af(bx) is a vertical stretch or compression by a factor of a and a horizontal compression or stretch by a factor of b. 
5/4f(x) is a vertical stretch by a factor of 5/4.
(-3, 8) translates to (-3, 8 • 5/4) = (-3, 10).

16. Work Session
Example 2. Point G is on the graph of f(x) = 3x, as shown. Determine the coordinates of point G under the transformation f(-4x).

17. Work Session
Example 2. Point G is on the graph of f(x) = 3x, as shown. Determine the coordinates of point G under the transformation f(-4x).
Solution: g(x) = af(bx) is a vertical stretch or compression by a factor of a and a horizontal compression or stretch by a factor of b. 
f(-4x) is a horizontal compression by a factor of 4, followed by a reflection across the y-axis since b is negative.
(2, 9) translates to (2 ÷ -4, 9) = (-1/2, 9).

18. Work Session
Example 3. The graph of f(x) = 2x is shown below. Determine the coordinates of point Q after the transformation f((2(x + 3)).

19. Work Session
Example 3. The graph of f(x) = 2x is shown below. Determine the coordinates of point Q after the transformation f((2(x + 3)).
Solution: g(x) = af(bx – h) + k is a vertical stretch or compression by a factor of a, a horizontal compression or stretch by a factor of b, a horizontal shift by h units and a vertical shift by k units. 
f ((2(x + 3)) is a horizontal compression by a factor of 2 followed by a horizontal translation 3 units to the left.
Perform the horizontal compression first:
(2, 4) translates to (2 ÷ 1/2, 4) = (1, 4).
Then, shift the point 3 units left:
(1,4) translates to (1 - 3, 4) = (-2, 4).

20. Work Session
Example 4. Describe the transformations of g(x) = 1/3 • 2x + 1 – 4 from f(x) = 2x.

21. Work Session
Example 4. Describe the transformations of g(x) = 1/3 • 2x + 1 – 4 from f(x) = 2x.
Solution: g(x) = af(bx – h) + k is a vertical stretch or compression by a factor of a, a horizontal compression or stretch by a factor of b, a horizontal shift by h units and a vertical shift by k units.
For g(x), a = 1/3, h = -1 and k = -4.
The graph is compressed vertically by a factor of 1/3, shifted 1 unit to the left, and 4 units down.

22. Homework
Complete:
Summary of the effects of the transformation af(bx – h) + k on f(x)

23. Work Session