Section 9.3 Day 1 & Day 2 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

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

African Mathematics Ndebele Houses

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?

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

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.

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

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 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

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

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

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

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

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

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

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

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 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

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 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 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: 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