Transformations of Quadratics and absolute value graphs

TRANSFORMATIONS OF QUADRATICS AND ABSOLUTE VALUE GRAPHS

Lesson 7.1

Recall Vertex Form of Quadratic Functions

𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 + 𝑘

This form can be very helpful for graphing other transformed quadratics without having to find the

x and y-intercepts.

Think about how “h” is the x-coordinate of the vertex.

• This tells us how much the graph shifts left or right (horizontal shift)

• Notice the – h in the formula. This always shifts it opposite.

Think about how “k” is the y-coordinate of the vertex.

• This tells us how much the graph moves up or down (vertical shift)

Now we will learn about how the graph of quadratic functions can be stretched or compressed.

Transformations of Quadratic Functions

This form allows stretches/compressions as well as the vertical and horizontal shifts.

𝑓 𝑥 = 𝑎 𝑏𝑥 − ℎ 2 + 𝑘

“a” – vertical stretch or compression and/or reflection

• This will affect our y-coordinates (think vertical)

• If a is negative, it will reflect across the x-axis

“b” – horizontal stretch or compression and/or reflection

• This will affect our x-coordinates by the reciprocal (think horizontal is opposite)

• If b is negative, it will reflect across the y-axis

Order of Transformations

When transforming graphs, you must transform in the following order:

1. Horizontal shifts (left and right)2. Stretches/compressions and Reflections3. Vertical shifts (up and down)

Example:

Tell what changes are made (in the correct order) to the graph of 𝑓 𝑥 = 𝑥2 to obtain each

graph:

1. 𝑓 𝑥 = − 𝑥 + 5 2 + 7 2. 𝑓 𝑥 = 3 𝑥 2 − 8 3. 𝑦 =1

2𝑥 + 1

2

Example:

Let 𝑓 𝑥 = 𝑥2, write a new function that translates 𝑓 𝑥 as described.

1. Vertical shrink of 1

3, left 5 units and up 2 units. 2. Horizontal stretch by 2, down 4 units

Transformations of Quadratic Functions

When transforming (translating) quadratic graphs, it is easiest to use the following special points:

0,0 1,1 (−1,1)(2,4)(−2,4)

Where do these points come from?

When we transform the quadratic equations, we will use these points and make the changes to

each x and y-coordinates.

Transformations of Quadratic Functions

𝑓 𝑥 = 𝑎 𝑏𝑥 − ℎ 2 + 𝑘

This is how we will change our points with the transformations:

“h” – horizontal shift

• Add or subtract to x-coordinates (opposite)

“a” – vertical stretch or compression

• Multiply to the y-coordinates

“b” – horizontal stretch or compression

• Multiply to the x-coordinates by the reciprocal (think horizontal is opposite)

“k” – vertical shift

• Add or subtract to y-coordinates

Example: Graph and label important points:

𝑓 𝑥 = 𝑥 + 1 2 + 3

Example: Graph and label important points:

𝑓 𝑥 = −2 𝑥 − 3 2

Example: Graph and label important points:

𝑓 𝑥 =1

3𝑥 + 4

2

− 2

Example: Change to vertex form, then draw the graph and label important points.

𝑓 𝑥 = −𝑥2 − 2𝑥 − 4

Transformations of Absolute Value Functions

𝑓 𝑥 = 𝑎 𝑏𝑥 − ℎ + 𝑘

This is just like the quadratic functions equation, except it contains and absolute value instead of a

square. The a, b, h, and k still transform the absolute value graphs in the same way.

However, since it is a different function, it has different points: 0,0 1,1 (−1,1)Where the points come from:

The shape is also different. Example of the shape:

Example:

Tell what changes are made (in the correct order) to the graph of 𝑓 𝑥 = 𝑥 to obtain each

graph:

1. 𝑓 𝑥 = −2 𝑥 + 1 − 2 2. 𝑓 𝑥 = −3𝑥 − 4

Example:

Let 𝑓 𝑥 = 𝑥 , write a new function that translates 𝑓 𝑥 as described.

1. Horizontal shrink of 3, left 2 units. 2. Vertical shrink of 1/2 , reflect y-axis, and up 1 unit

Example: Graph and label important points:

𝑓 𝑥 = 3 𝑥 + 3

Example: Graph and label important points:

𝑓 𝑥 = −1

2𝑥 − 2 + 3

ASSIGNMENT 7.1

Problems #1-25 from the packet

BRING UNIT 2 WORKBOOKS NEXT TIME

(You will need them for the rest of the unit)