Algebra 2A Unit 3 Portfolio Translating and Scaling Functions

Prior Knowledge Questions (Do these BEFORE using the Gizmo.)

Shown below are the graphs of two functions, y = f ( x ) and y = g ( x ).

y = f(x) y = g(x)

1. How would you have to "move" the graph of y = f(x) so that it would match 1. _____

the graph of y = g(x)?

A. Shift y = f(x) left 4 units. B. Shift y = f(x) right 4 units. C. Shift y = f(x) up 4 units. D. Shift y = f(x) down 4 units

2. The point (0, 0) lies on the graph of y = f ( x ). What are the coordinates of 2. _____

the point on y = g(x) to which the point (0, 0) was translated?

A. (-4, 0) B. (0, 4) C. (4, 0) D. (0, -4)

3. With Show parent function selected, vary h with the slider. 3. _____

What happens to the graph?

A. The graph shifts up and down (vertically). B. The graph gets wider (expands). C. The graph gets more narrow (contracts). D. The graph shifts left and right (horizontally).

4. Drag the k slider back and forth. How does the graph change as you vary k ? 4. _____

A. The graph shifts up and down (vertically). B. The graph gets wider (expands). C. The graph gets more narrow (contracts). D. The graph shifts left and right (horizontally).

Activity A:

Effects of h and k on the graph

Get the Gizmo ready :

Be sure the TRANSLATION tab, abs. value function, and Show parent function are selected.

Set h to 0 and k to 0 so that f ( x ) = | x | is graphed.

Keeping h = 0, set k to 3 to graph y = f ( x ) + 3, shown in red. (To set a slider to a specific value, click on the number in the text field, type the new value and hit Enter .)

5. How does the graph change? 5. _____

A. The graph shifts up 3 units. B. The graph shifts down 3 units. C. The graph shifts right 3 units. D. The graph shifts left 3 units.

6. On the blue parent function graph, the coordinates of the vertex are (0, 0). 6. _____

What is the vertex of the red translated (shifted) graph?

A. (3, 0) B. (-3, 0) C. (0, 3) D. (0, -3)

7. The ordered pair (2, 2) lies on the graph of the parent function. What point 7. _____

must lie on the graph of y = f ( x ) + 3? Hint: A change in the k value will only affect the y-coordinate.

A. (2, -3) B. (2, 3) C. (2, -5) D. (2, 5)

8. Use the sliders to vary the values of h and k . Watch the graph as you do. 8. _____

In general, how do the values of h and k relate to the vertex of the graph?

A. The vertex of the graph is always (- h, - k). B. The vertex of the graph is always (- h, k) C. The vertex of the graph is always (h, - k). D. The vertex of the graph is always (h, k).

With abs. value selected, set h to 2 and k to 0 to graph y = f ( x - 2), shown in red.

9. What is the vertex of the original (parent) graph in blue? 9. _____

A. (0, 0) B. (-2, 0) C. (0, -2) D. (-2, -2)

10. What is the vertex of the new graph? 10. _____

A. (0, 0) B. (2, 0) C. (0, 2) D. (-2, -2)

11. How was the graph translated? 11. _____

A. The graph shifted down 2 units. B. The graph shifted up 2 units. C. The graph shifted left 2 units. D. The graph shifted right 2 units.

12. If the point (3, 3) lies on the graph of y = f ( x ), what point lies on y = f ( x - 2)? 12. _____

Hint: What value of x will make (x - 2) equal 3?

A. (5, 3) B. (5, -3) C. (-5, 3) D. (-5, -3)

13. How do you think the graph of y = f ( x - h ) relates to the graph of y = f ( x )? 13. _____

A. It shifts h units to the left. B. It shifts h units to the right. C. It shifts h units up. D. It shifts h units down.

Select parabola so that f ( x ) = x 2 is the parent function. Set h to -5 to graph y = f ( x + 5).

14. How has the graph been translated? 14. _____

A. The graph shifts up 5 units. B. The graph shifts down 5 units. C. The graph shifts right 5 units. D. The graph shifts left 5 units.

15. What value of x makes f(x + 5) equal f(0)? 15. _____

A. x = 5 B. x = - 5 C. x = 0 D. There are no values of x that will make f(x + 5) equal f(0).

16. If (2, 4) lies on the graph of y = f(x), what point must lie on y = f(x + 5)? 16. _____

A. (7, 4) B. (-3, 4) C. (7, 9) D. (2, 9)

17. If ( x , y ) lies on the graph of y = f ( x ), what point must lie on y = f ( x + h )? 17. _____

Hint: A change in the h value will only affect the x-coordinate.

A. (x + h, y) B. (x, y + h) C. (x - h, y) D. (x, y - h)

The graph of an absolute value function is shown below.

18. What is the vertex? 18. _____

A. ((0, 4) B. (4, 0) C. (-4, 0) D. (0, -4)

19. If f(x) = , what function is graphed here? 19. _____ x| |

A. f(x - 4) B. f(x + 4) C. f(x) - 4 D. f(x) + 4

Now select Show parent function and click on parabola so the parent function is f ( x ) = x 2 .

20. Vary the a slider. What happens to the graph as you do so? 20. _____

A. As a increases, the graph gets less steep and as a decreases, the graph gets steeper. B. As a increases, the graph shifts up and as a decreases, the graph shifts down. C. As a increases, the graph shifts right and as a decreases, the graph shifts left. D. As a increases, the graph gets steeper and as a decreases, the graph gets less steep.

Now set a = 2 to graph y = 2 f ( x ).

21. The point (-2, 4) lies on the graph of the parent function y = f ( x ). What point 21. _____

must lie on the graph of y = 2 f ( x )? Hint: The x-coordinate will be the same but the y-coordinate will change.

A. (-2, 2) B. (-2, 6) C. (-2, 8) D. There is not enough information given to determine a point on y = 2f(x)?

The value of a scales (stretches or shrinks) the graph vertically.

Activity B:

Practice scaling and translating functions

Get the Gizmo ready :

Select the BOTH tab.

Unselect Show parent function .

22. How has the parent function ( f ( x ) = | x |) been transformed to create 22. _____

the graph shown? Hint: The vertex of the graph shown is (3, -4)

A. The graph has been shifted to the right 4 and down 3 . B. The graph has been shifted to the right 3 and up 4. C. The graph has been shifted to the right 3 and down 4. D. The graph has been shifted to the left 3 and down 4.

23. What are the values of h and k? 23. _____

A. h = 3 and k = - 4 B. h = -3 and k = - 4 C. h = 3 and k = 4 D. h = - 3 and k = 4

24. In y = f ( x ) notation, write the function graphed above. 24. _____

Hint: Using the Gizmo, check your answers.

A. y = f(x + 3) + 4 B. y = f(x - 3) + 4 C. y = f(x + 3) - 4 D. y = f(x - 3) - 4

A parent function g ( x ) = x 2 and a transformed graph are shown below.

25. Describe how the graph was transformed and state the values of h and k. 25. _____

A. The graph was shifted right 4 and down 2. h = -4 and k = 2 B. The graph was shifted left 4 and up 2. h = -4 and k = 2 C. The graph was shifted left 4 and up 2. h = 4 and k = 2 D. The graph was shifted left 4 and up 2. h = 2 and k = -4

26, I n y = g ( x ) notation, write the function graphed above. 26. _____

Hint: In the Gizmo, select parabola and check your answers.

A. y = f(x - 4) + 2 B. y = f(x + 4) + 2 C. y = f(x - 4) - 2 D. y = f(x + 4) - 2

The point (4, 16) lies on the graph of the parent function f ( x ) = x 2 .

27. If the point (1, 16) lies on the graph of y = f ( bx ), what is the value of b ? 27. _____

Hint: What value of b will give you back the y-coordinate of 16?

A. b = 1 B. b = 2 C. b = 4 D. b = 16

28. What point on the graph of y = 0.5 f ( x ) corresponds to (4, 16)? 28. _____

Hint: The function multiplies the y-value by 0.5.

A. (4, 0.5) B. (4, 8) C. (2, 0.5) D. (2, 8)

29. How do the graphs of the functions y = f ( x ) and y = f (- x ) relate when 29. _____

f ( x ) = x 2 ?

A. The graphs reflect each other. B. The second graph is a horizontal shift of the first graph by x units. C. The graphs are identical. D. The second graph is a vertical shift of the first graph by x units.

30. Write the equation of the function g(x) that translates the parent function 30. _____

f(x) = x2 by shifting g(x) 7 units to the left and 5 units down.

A. g(x) = (x - 7)2 - 5 B. g(x) = (x + 7)2 - 5 C. g(x) = (x - 5)2 - 7 D. g(x) = (x + 5)2 - 7

Respond to each of the following questions. Be sure to include and thoroughly describe each transformation.

31. Describe the transformations necessary to transform the graph of

f(x) to that of g(x). Hint: Be sure to include and thoroughly describe each transformation.

You may use the Gizmo to check your answer.

f(x) = x2 and g(x) = 3(x - 1)2 + 4

32. Given the parent function f(x) = x2, define the function g(x) that

transforms f(x) as follows:

- Reflection over the x-axis - Horizontal shift left 3 units - Vertical shift up 7 units

Hint: You may use the Gizmo to check your answer.