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Section 9.3 Day 1 Transformations of Quadratic Functions Algebra 1

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

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.

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

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

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

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

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

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

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

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

Stretched Vertically: 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

Compressed Vertically: 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

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

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

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

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

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

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

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

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

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

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

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

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

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