Warm-Up. Transformations of Quadratic Functions. Recalling Slope-Intercept Form. y 4. Table of values Standard Form Slope-Intercept Form

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Warm-Up Transformations of Quadratic Functions

Recalling Slope-Intercept Form

𝑥 y

4

–4

–4 4

−3𝑥 + 𝑦 = 1

x y

4

–4

–4 4

𝑦 = 3𝑥 + 1

run = 1 rise = 3

𝒙 𝒚

–3 –8

–1 –2

1 4

3 10

Table of values Standard Form Slope-Intercept Form

Slope-intercept form:

x y

4

–4

–4 4

𝑦 = 𝑥

𝑦 = 𝑚𝑥 + 𝑏

x y

4

–4

–4 4

𝑦 = 3𝑥 + 1

run = 1 rise = 3

𝑦 =

1𝑥 + 0

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Warm-Up Transformations of Quadratic Functions

Lesson Question

Lesson Objectives

By the end of this lesson, you should be able to:

• Use completing the to write quadratic functions in the form .

• Describe the effects of changes in 𝑎, ℎ, and 𝑘 to the of a function in the form 𝑦 = 𝑎 𝑥 − ℎ ²+ 𝑘.

graph square

𝑦 = 𝑎 𝑥 − ℎ ²+ 𝑘

transformation

an operation on a figure that may change its location in a plane, , or shape

vertex form of a quadratic equation

the form , where 𝑎, ℎ, and 𝑘 are real numbers, and 𝑎 is not , and in which the point (ℎ, 𝑘) is the of the parabola

Words to Know

Fill in this table as you work through the lesson. You may also use the glossary to help you.

W

2

K

orientation

zero

𝑦 = 𝑎 𝑥 − ℎ ²+ 𝑘

vertex

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Instruction Transformations of Quadratic Functions

Lesson Question

Can graphing a quadratic function be made easier by changing its form?

A transformation of an equation is an

operation on an equation that may change its in a plane or its .

• Translation (slide)

• Reflection (flip)

•

• Stretch or Shrink

Transformations

?

Slide

2 Vertical Translation of 𝒚 = 𝒙

^𝟐

𝑦 = 𝑎𝑥²+ 𝑘, 𝑎 = 1

Because 𝑘 is added to the -values of 𝑦 = 𝑥², the 𝑦-values increase and decrease, causing vertical shifts.

• If 𝑘 > 0, the graph shifts 𝑘 units .

• If 𝑘 < 0, the graph shifts 𝑘 units .

location

downward

shape

Dilation

x y

2

upward

^x

y

4

–4 4

8

𝑦 = 𝑥²+ 2 𝑦 = 𝑥²

𝑦 = 𝑥²− 1

4

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Instruction

6

Slide

Transformations of Quadratic Functions

Horizontal Translation of 𝒚 = 𝒙

^𝟐

Vertex Form

The vertex form of a quadratic equation is the form 𝑦 = 𝑎 𝑥 − ℎ ²+ 𝑘, where:

• 𝑎, ℎ, and 𝑘 are real numbers.

• 𝑎 is not .

• (ℎ, 𝑘) is the of the parabola.

𝑦 = 𝑎 𝑥 − ℎ ², 𝑎 = 1

Because ℎ is subtracted from , the same graph appears when 𝑥 is ℎ units

greater.

• If ℎ > 0, the graph shifts ℎ units to the .

• If ℎ < 0, the graph shifts ℎ units to the .

left

right

^x

y

–4 2

8

–2 6 4

𝑦 = 𝑥² 𝑦 = (𝑥 + 2)²

𝑦 = (𝑥 − 3)²

zero

vertex

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x y

–2 2

4 2

–4 –2

–4 4

𝑦 =

𝑦 = 𝑎𝑥²

x y

–2 2

4 2

–2

–4 4

𝑦 = + 1

𝑦 = (𝑥 − 3)²+ 1

–4

Instruction

6

Slide

Transformations of Quadratic Functions

Quadratic Function Equations

Example:

How 𝒂 Affects the Orientation of a Parabola

𝑥 − 2 − 2 + 1

𝑦 = 𝑥²− 4𝑥 + 4 + 1 𝑦 = 𝑥²− 4𝑥 + 5

x –1 0 2 4 3

y 10 5 1 5 10

x y

–2 2

8 6 4 2 10

2

_,

1

9

What happens if we change the value of 𝑎 from positive to negative?

• The orientation changes (flips upside down).

• The vertex (ℎ, 𝑘) stays the same.

• The parabola is reflected over the line 𝑦 = 𝑘.

−

²

−( − 3)

²

𝑦 = 𝑥 − 2 ²+ 1 𝑦 =

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Instruction

12

Slide

Transformations of Quadratic Functions

The Effect of 𝒂 on 𝒚 = 𝒂𝒙

^𝟐

Case Effect

𝑎 > 1 The parabola stretches .

𝑎 < −1 The parabola flips and vertically.

−1 < 𝑎 < −1 The parabola vertically (widens).

x y

–2 2

4 2

–4 –2

–4 4

𝒂 is sometimes called the “stretch factor” of a parabola.

vertically

shrinks

𝑗 𝑥 = −4𝑥² ℎ 𝑥 = 0.5𝑥² 𝑔 𝑥 = 2𝑥² 𝑓 𝑥 = 𝑥²

𝑘 𝑥 = −1 6𝑥²

stretches

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Instruction

14

Slide

Transformations of Quadratic Functions

Identifying the Constants in Vertex Form

Vertex form: 𝑦 = 𝑎 𝑥 − ℎ ²+ 𝑘 Example: 𝑦 = 5 𝑥 − 7 ²+ 9

ℎ = 7 𝑘 = 9 𝑎 =

vertex at

7

, 9

5

Example: 𝑦 = − 𝑥 + 3 ² − 4 𝑎 =

ℎ = −3 𝑘 = −4

vertex at −3,

−1

−4

x 2

10

6 2

14

10 18

6

14 18

(7, 9) y

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Instruction

17

Slide

Transformations of Quadratic Functions

Completing the Square to Change a Quadratic Function to Vertex Form

Example:

Changing a Quadratic Function to Vertex Form When 𝒂 ≠ 𝟏

𝑦 = 𝑥²− 4𝑥 + 5

𝑦 = 𝑥²− 4𝑥 + −2 ²+ 5 − −2 ² 𝑦 = 𝑥²− 4𝑥 + 4 + 5 − 4

𝑦 = (𝑥 − 2) 𝑥 − 2 + 1 𝑦 =

(𝑥 − 2)

²

+ 1

Write the function in standard form.

Add the square of of𝑏.

Also, subtract it.

Factor the perfect-square

trinomial.

Write the function in form.

half

vertex

Example:

𝑦 = 4𝑥²− 8𝑥 + 5 𝑦 = 4(𝑥²− 2𝑥) + 5

𝑦 = 4 𝑥²− 2𝑥 + −1 ² + 5 − 4 −1 ² 𝑦 = 4(𝑥 − 1) 𝑥 − 1 + 5 − 4

Write in form.

Factor 4 from the first two terms.

Add and subtract so the function remains . Factor the perfect-square

standard

unchanged

19

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Summary

Answer

Can graphing a quadratic function be made easier by changing its form?

Transformations of Quadratic Functions

Lesson Question

?

Slide

2 Review: Key Concepts

(Sample answer) Graphing a quadratic function is made easier by writing the function in the vertex form, 𝑦 = 𝑎 𝑥 − ℎ

²

+ 𝑘. In this equation, indicates whether the parabola opens up or down, and its magnitude affects the width of the parabola. The vertex of the parabola is ( ).

• The vertex form of a quadratic function is 𝑦 = 𝑎 𝑥 − ℎ ²+ 𝑘.

• The of its graph is (ℎ, 𝑘).

• The constant a indicates

whether the parabola opens up or down, and its magnitude affects the of the parabola.

vertex

width

𝑓(𝑥) = 𝑥² 𝑓(𝑥) = 𝑥 − ℎ ²+ 𝑘

(h, k)

(0, 0)

h k

2 6 10

2 6

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Slide

Review: Common Problem Types

Summary

2 Transformations of Quadratic Functions

To write a quadratic function in vertex form:

1. Write the equation in standard form: 𝑦 = 𝑎𝑥²+ 𝑏𝑥 + 𝑐.

2. If 𝑎 is not 1, it out of the first two terms.

3. Form 𝑎 perfect-square trinomial by taking of 𝑏 and squaring it, then adding it to the first two terms and subtracting it from the constant.

4. Write the trinomial as a binomial . 5. Write it in vertex form, 𝑦 = 𝑎 𝑥 − ℎ ²+ 𝑘.