1. Section 9.3 Day 1 & Day 2 Transformations of Quadratic Functions
Algebra 1

2. Learning Targets
Define transformation, translation, dilation, and reflection
Identify the parent graph of a quadratic function
Graph a vertical and horizontal translation of the quadratic graph
Describe how the leading coefficient of a quadratic graph changes the dilation of the parent graph
Graph a reflection of a quadratic graph
Identify the quadratic equation from a graph

3. Key Terms and Definitions
Transformation:
Changes the position or size of a figure
Translation:
A specific type of transformation that moves a figure up, down, left, or right.

4. Key Terms and Definitions
Dilation:
A specific type of transformation that makes the graph narrower or wider than the parent graph
Reflection:
Flips a figure across a line

5. African Mathematics Ndebele People of South Africa
One of the Nguni Tribes
2/3 of South Africa’s African population
Art is a significant part of the culture
Reinforce the Ndebele identity
Ndebele house paintings
Represented the harsh times post Boer wars

6. African Mathematics
Ndebele Houses

7. African Mathematics Ndebele Beadwork
Identify some of the transformations in the strip beadworks.
Horizontal Reflection, Vertical Reflection, Translation,
Which one is not like the others?

8. Key Terms and Definitions
Parent Quadratic Graph:
𝑓 𝑥 = 𝑥 2
General Vertex Form:
𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘

9. Instructions Exploration in Groups
Each person should have a piece of graph paper
Each person will graph the “Entire Group” equation and their respective color on their graph.
Then, individuals will share out the ordered pairs to each other to graph each other’s equations.
Thus, each side of graph paper should have 5 different colored equations.
Once you have graphed all of the equations, your task is to determine how the components of the equation impact the visual representation of the graph.

10. Side 1: Side 2: Exploration in Groups Entire Group: 𝑦= 𝑥 2 +2
Green: 𝑦= 𝑥
Orange: 𝑦= 𝑥−
Pink: 𝑦= 𝑥−
Yellow: 𝑦= 𝑥
Side 2:
Entire Group: 𝑦= 𝑥−1 2
Green: 𝑦= 𝑥−
Orange: 𝑦= 𝑥−1 2 −3
Pink: 𝑦= 𝑥−
Yellow: 𝑦= 𝑥−1 2 −4

11. Horizontal Translations
In vertex form 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘, ℎ represents the horizontal translation from the parent graph.
To the Right: 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘 of ℎ units
To the Left: 𝑓 𝑥 =𝑎 𝑥+ℎ 2 +𝑘 of ℎ units

12. Vertical Translations
In vertex form 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘, 𝑘 represents the vertical translation from the parent graph.
Upward: 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘 of 𝑘 units
Downward: 𝑓 𝑥 =𝑎 𝑥−ℎ 2 −𝑘 of 𝑘 units

13. Horizontal Translations: Example 1
Given the function 𝑓 𝑥 = 𝑥−2 2
A) Describe how the function relates to the parent graph
Horizontal shift 2 units to the right
B) Graph a sketch of the function

14. Vertical Translations: Example 2
Given the function ℎ 𝑥 = 𝑥 2 +3
A) Describe how the function relates to the parent graph
Vertical shift 3 units upward
B) Graph a sketch of the function

15. Translations: Example 3
Given the function 𝑔 𝑥 = 𝑥+1 2
A) Describe how the function relates to the parent graph
Horizontal Shift one unit to the left
B) Graph a sketch of the function

16. Translations: Example 4
Given the function 𝑔 𝑥 = 𝑥 2 −4
A) Describe how the function relates to the parent graph
Vertical shift 4 units downward
B) Graph a sketch of the function

17. Exit Ticket for Feedback
Using your knowledge of horizontal and vertical translations, determine the equation of the graph in vertex form.

18. Dilations
In vertex form 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘, 𝑎 represents a dilation from the parent graph.
Compressed Horizontally: 𝑎 >1
Stretched Horizontally: 0< 𝑎 <1

19. Stretched Horizontally: Example 1
Given the function ℎ 𝑥 = 1 2 𝑥 2
A) Describe how the function relates to the parent graph
Stretched Horizontally
B) Graph a sketch of the function

20. Compressed Horizontally: Example 2
Given the function 𝑔 𝑥 =3 𝑥 2 +2
A) Describe how the function relates to the parent graph
Compressed Horizontally
Vertical Shift 2 units upward
B) Graph a sketch of the function

21. Dilation: Example 3 Given the function 𝑗 𝑥 = 1 3 𝑥 2 +2
A) Describe how the function relates to the parent graph
Stretched Horizontally
Vertical Shift 2 units upward
B) Graph a sketch of the function

22. Dilation: Example 4 Given the function 𝑘 𝑥 =2 𝑥 2 −12
A) Describe how the function relates to the parent graph
Compressed Horizontally
Vertical Shift 12 units downward
B) Graph a sketch of the function

23. Reflections Across the x-axis
In vertex form 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘, the sign of 𝑎 can represent a reflection across the x-axis from the parent graph.
In particular, 𝑎 is positive there is no change.
If 𝑎 is negative, there is a flip across the x-axis

24. Reflections Across the y-axis
In vertex form 𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘, the sign of 𝑥 can represent a reflection across the y-axis from the parent graph.
In particular, 𝑥 must be negative

25. Reflections Across the x-axis: Example 1
Given the function 𝑔 𝑥 =− 𝑥 2 −3
A) Describe how the function relates to the parent graph
Reflection across the x-axis
Vertical Shift 3 units downward
B) Graph a sketch of the function

26. Reflections Across the y-axis: Example 2
Given the function 𝑓 𝑥 =2 −𝑥 2 −9
A) Describe how the function relates to the parent function
Reflection across the y-axis
Vertical Shift 9 units downward
B) Graph a sketch of the function

27. Reflections: Example 3 Given ℎ 𝑥 =−2 𝑥−1 2
A) Describe how the function relates to the parent graph
Reflection across the x-axis
Compressed Horizontally
Horizontal Shift one unit to the right
B) Graph a sketch of the function

28. Reflections: Example 4 Given the function 𝑔 𝑥 = −𝑥 2 +2
A) Describe how the function relates to the parent graph
Reflection across the y-axis
Vertical shift 2 units upward
B) Graph a sketch of the function

29. Identifying From a Graph Procedure
1. Check for a horizontal or vertical translation
2. Check for a reflection across the x-axis
3. Check for a dilation

30. Identifying: Example 1
Which is an equation for the function shown in the graph?
A) 𝑦= 1 2 𝑥 2 −5
B) 𝑦=− 1 2 𝑥 2 +5
C) 𝑦=−2 𝑥 2 −5
D) 𝑦=2 𝑥 2 +5

31. Identifying: Example 3
Which equation is shown for the function in the graph?
A) 𝑦=−3 𝑥 2 +2
B) 𝑦=3 𝑥 2 +2
C) 𝑦=− 1 3 𝑥 2 −2
D) 𝑦=− 1 3 𝑥 2 −2

32. Identifying: example 4
Which is an equation for the function shown in the graph?
A) 𝑦= 𝑥
B) 𝑦= 𝑥−
C) 𝑦=− 𝑥+2 2 −2
D) 𝑦=− 𝑥−2 2 −2