math-2 lesson 2-2 the quadratic function and how it is...

Math-2Lesson 2-2

The Quadratic Function and How It is Transformed

VocabularyTransformation: an adjustment made to the parent function

that results in a change to the graph of the parent function.

Changes could include:

shifting (“translating”) the graph up or down,

“translating” the graph left or right

vertical stretching or shrinking

Reflecting across x-axis or y-axis

Graphical TransformationsParent Function: The simplest function in a family

of functions (lines, parabolas, cubic functions, etc.)

2xy

Vertex: The point where

the graph stops going

down and starts going

back up.

Where is the vertex of the

parent function?

(0, 0)

2)( 2 xxg2)( xxf

Build a table of values for each equation for domain

elements: -2, -1, 0, 1, 2.

Why does adding 2 to the

parent function translate

the graph up by 2?

x f(x)

-2

-1

0

1

2

4

1

0

1

4

x g(x)

-2

-1

0

1

2

6

3

2

3

6

2 has been added to

the output of the

parent function.

22

2 xy2

1 xy Notice the subscripts.

Why did I put different

subscripts on the two

equations?

shows they are different equations.

easier to see in function form with different names2

1 xy 2)( xxf

22

2 xy

2)( 2 xxg

2)()( xfxg

Take Away: add 2 to the parent function move up 2

Your Turn:Describe the transformation to the parent

function:

2xy

42 xy

Describe the transformation to the parent

function:

2xy

52 xy

translated down 4

translated up 5

2)( xxf

Multiplying the parent function by 3, makes it look “steeper”

23)( xxg

2)( xxf Why does multiplying

the parent function by 3

cause the parent to

look steeper?

23)( xxg

Build a table of values for each equation for some of

the input values: -2, -1, 0, 1, 2.

x f(x)

-2

-1

0

1

2

x g(x)

-2

-1

0

1

2

4

1

0

1

4

12

3

0

3

12

Same input value

output value has been

multiplied by 3.

We say the

function has been

“vertically

stretched” by a

factor of 3.

2)( xxf Multiplying the parent

function by -1, reflects

across the x-axis.

2)( xxg

2)()( xxfxg

x f(x)

-2

-1

0

1

2

4

1

0

1

4

x -f(x)

-2

-1

0

1

2

Multiplying the parent

function by -1, multiplies

each y-value by -1.

-4

-1

0

-1

-4

2)( xxf 2)1()( xxg

Vertex: (1, 0)

If you move the whole graph right 1, which

coordinate of the vertex changes, the x-coordinate

or the y-coordinate?

Vertex: (0, 0)

g(1) = ?

2)11()( xg 0

2)( xxf

Build a table of values for each equation for domain

elements: -2, -1, 0, 1, 2.

x f(x)

-2

-1

0

1

2

x g(x)

-2

-1

0

1

2

4

1

0

1

4

9

4

1

0

1

2)1()( xxg

Replacing ‘x’ in the original function with ‘x – 1’

causes the graph to translate right ‘1’

These effects accumulate


function:

2)( xxf

2)( 2 xxg

2)( 2 xxg

Reflected across x-axis and translated up 2

2)()( xfxg

Describe graphically how f(x) is transformed to get g(x).

These effects accumulate

2)( xxf

Describe the algebraic transformation to the

parent function:

Multiplying the parent function by 3 then subtracting 6…

63)( 2 xxg

6)(3)( xfxg

Transformations: (1) vertically stretched by 3,

(2) Down 6.

2)( xxf

Which transformation is it?

Up 5

2)5()( xxk

5)( 2 xxg

Where is new vertex?

(x, y) = (0, 5)

How could you “test”

each function to see if it

is the correct one?

??? 5)0( g

??? 5)0( k

5)( 2 xxg

5)0()0( 2 g2)50()0( k

5) (up 5)( 2 xxg

5)(left )5()( 2 xxk

Let’s generalize the transformations

Reflection

across x-axis

translating

up or downvertical

stretch

factor

khxay 2)()1(

Translates

left/right

4)3(2 2 xy

Reflected across x-axis, twice as steep,

translated up 4, translated right 3

2)( xxf

Your Turn:


function:2)( xxf

3)5( 2 xy

translated up 3

translated left 5

Your Turn:


function:2xy 2)1(2 xy

Vertically stretched by a

factor of 2, translated right 1

Your Turn:


function: 2xy 4)3(2

1 2 xy

Reflected across x-axis

Vertically stretched by a factor of ½

(shrunk), translated up 4

translated left 3

Your Turn:


function: 2xy 54 2 xy

Vertically stretched by a factor of 4, down 5

VSF = 4, down 5

How can you tell if the graph has been vertically stretched?

Right 1

Up 1

2xy 22xy

Right 1

+2

Find the vertex. Go right 1, then count how many space you have to go up to get to the graph.

How can you tell what the left/right and up/down shifts are?

2xy

2)3()( 2 xxg

Right 3

up 2

Count how far you have to move right/left from (0, 0) and then up/down to get to the new vertex.

Test your equation!

??? 3)4( g

(3, 2)

2)34()4( 2 g

yes! 3)4( g

(4, 3)

What is the equation of the graph?

2xy

4)1()( 2 xxg

Left 1

Up 4

1. Find left/right and up/down shift.

Test your equation! g(0) = 1 ??

(-1, 4)

4)1(3)( 2 xxg

2. Find reflection and vertical stretch.

right 1

- 3

4)10(3)0( 2 g

yes! 1)0( g

math-2 lesson 2-2 the quadratic function and how it is...

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