Module 19.2

Transforming Quadratic Functions

P. 903

How can you obtain the graph of 𝒈 𝒙 = 𝒂(𝒙 − 𝒉)𝟐+𝒌from the graph of 𝒇 𝒙 = 𝒙𝟐?

P. 904

We know what 𝑓 𝑥 = 𝑥2 looks like. But what does the function look like when it is shifted up or down?

P. 904

P. 905

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P. 906

We know what 𝑓 𝑥 = 𝑥2 looks like. But what does the function look like when it is shifted left or right?

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P. 907

Now put the a, h, and k together.

P. 908

Quadratic functions can take three forms – Standard, Vertex, and Factored.In Module 19.1 we mentioned Standard Form: 𝒇 𝒙 = 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄Here’s Vertex Form:

𝒈 𝒙 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌

Width Horizontal Translation

Vertical Translation

Sign

The point (h,k) is the vertex.The Axis of Symmetry runs through 𝒉.So the equation for that line is 𝒙 = 𝒉.

Example: 𝒈 𝒙 = 𝟑(𝒙 − 𝟐)𝟐 + 𝟒In this case, the vertex (h,k) = (2,4).Example: 𝒈 𝒙 = −𝟑 𝒙 + 𝟏 𝟐 − 𝟑In this case, the vertex (h,k) = (–1, –3).

To graph a function in this form:

1) Identify the vertex.2) Generate two points on either side of the vertex.3) Draw the parabola!

P. 908

Helpful: Determine from a whether the parabola should open upward or downward. Does it match your graph?

P. 908

Vertex

Horizontal

VerticalVertex

P. 908

Vertex

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P. 909

P. 909

𝐓𝐡𝐞 𝐯𝐞𝐫𝐭𝐞𝐱 𝐢𝐬 𝐚𝐭 𝟐, 𝟒 .

P. 909

P. 909

𝐓𝐡𝐞 𝐯𝐞𝐫𝐭𝐞𝐱 𝐢𝐬 𝐚𝐭 −𝟑,−𝟏 .