1. Section 9.3 Day 1 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. Key Terms and Definitions
Parent Quadratic Graph:
𝑓 𝑥 = 𝑥 2
General Vertex Form:
𝑓 𝑥 =𝑎 𝑥−ℎ 2 +𝑘

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

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

13. Vertical Compression: Example 1
Given the function ℎ 𝑥 = 1 2 𝑥 2
A) Describe how the function relates to the parent graph
Compressed Vertically (Wider)
B) Graph a sketch of the function

14. Vertical Stretch: Example 2
Given the function 𝑔 𝑥 =3 𝑥 2 +2
A) Describe how the function relates to the parent graph
Stretched Vertically (Narrower)
Vertical Shift 2 units upward
B) Graph a sketch of the function

15. Dilation: Example 3 Given the function 𝑗 𝑥 = 1 3 𝑥 2 +2
A) Describe how the function relates to the parent graph
Compressed Vertically (Wider)
Vertical Shift 2 units upward
B) Graph a sketch of the function

16. Dilation: Example 4 Given the function 𝑘 𝑥 =2 𝑥 2 −12
A) Describe how the function relates to the parent graph
Stretched Vertically (Narrower)
Vertical Shift 12 units downward
B) Graph a sketch of the function

17. 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 must be a negative number

18. 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

19. 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
Stretched Vertically
Horizontal Shift one unit to the right
B) Graph a sketch of the function

20. Identifying From a Graph Procedure
Check for a horizontal or vertical translation
Check for a reflection across the x-axis
Check for a dilation

21. 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

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

23. 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

24. 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