math-3 lesson 1-3 quadratic function and how it is...

Math-3 Lesson 1-3

Quadratic Function and How It is Transformed

Vocabulary Transformation: 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

horizontal stretching or shrinking

Squaring Function

2)( xxf

Graphical Transformations Parent Function: The simplest function in a family

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

2xy

2xy

Replacing ‘y’ with y – 2 results in an up two translation.

22 xy

22 xyWe usually write

“y as a function of x”

“y” on one side of “=“ sign,

operations performed on ‘x’ on the other side of the “=“ sign

Adding ‘2’ to the parent function translates the parent

function up two.

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

For each respective

input value, the

output value has

increased by 2.

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 names 2

1 xy 2)( xxf

22

2 xy

2)( 2 xxg

2)()( xfxg

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

If you add 2

to any function that function

will translate upward 2 units.

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.

f(x) is given in

the table below

x f(x)

-2 -2

-1 0

0 2

1 4

2 6

g(x) = 3*f(x)

Fill in the table below

x g(x)

-2

-1

0

1

2

-6

0

6

12

18

2)( xxf

Multiplying the x-value by 2,

horizontally shrinks the graph.

2)2()( xxg 2)2()2()( xxfxg

x f(x)

-2

-1

0

1

2

4

1

0

1

4

x f(2x)

-2

-1

0

1

2

Looks like a vertical stretch!

16

4

0

4

16

22 4)2()( xxxg

For the square function:

Horizontal shrinking

Looks like

Vertical stretching.

2)( xxf 22 4)2()( xxxg

For the square function:

Horizontal stretch by ½

(multiply x-value of point by ½

Looks like

Vertical stretch by 4

(multiply y-value of point by 4).

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

Remember? Replacing ‘y’ with ‘y-1’ moved it up 1.

Replacing ‘x’ with ‘x-1’ moves it right 1.

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

)()1()1( 2 xgxxf

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

2)()( xfxg

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

2)( xxf

)1(* )1(* 2)( xxf

2 2

22)( 2 xxf



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).


2)( xxf

Describe the algebraic transformation to the

parent function:

Multiplying the parent function by 3 then subtracting 6…

63)( 2 xxg6)(3)( xfxg

2)( xxf

3* 3*

23)(3 xxf

6 6

636)(3 2 xxf


2)( xxf

Describe the graphical transformation to the

parent function:

Multiplying the parent function by 3 then

subtracting 6…

Vertically stretched by a factor of 3 and

translated down 6

63)( 2 xxg

6)(3)( xfxg

Let’s generalize the transformations

Reflection

across x-axis

translating

up or down vertical

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

3)5(3)5( 22 xxf

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 5)2( 2 xy

Horizontally shrunk by 2 (or stretched by ½)

translated up 5

Your Turn: Describe how each x-y pair is changed:

3)4(2)( xfxg

x f(x)

-2

-1

0

1

2

4

2

0

-3

-5

x -2f(x)

-2

-1

0

1

2

-8

-4

0

6

10

x -2f(x-4)

-8

-4

0

6

10

2

3

4

5

6

x -2f(x-4)+3

-5

-1

3

9

13

2

3

4

5

6

math-3 lesson 1-3 quadratic function and how it is...

Documents