Math-3 Lesson 1-3

Quadratic, Absolute Value and Square Root Functions

5432 rr

442 2 xx

Graphical TransformationsTransformation: an adjustment made to the parent function that results in a change to the graph of the parent function.

Changes could include:shiftingshifting (“translating”) the graph (“translating”) the graph upup or or downdown,,

““translating”translating” the graph the graph leftleft or or rightright

vertical stretchingvertical stretching or or shrinkingshrinking(making the graph (making the graph more steep more steep or or less steepless steep) )

ReflectingReflecting across x-axis or y-axis across x-axis or y-axis

Graphical TransformationsParent Function: The simplest function in a familyof functions (lines, parabolas, cubic functions, etc.)

2xy

Compare the two parabolas.

22 xy

2xy

Build a table of values for each equation for domain elements: -2, -1, 0, 1, 2.

Why does adding 2 to the parent function shift the graph up by 2?

22 xy

x y

-2

-1

0

1

2

41014

x y

-2

-1

0

1

2

63236

Your Turn:Describe the transformation to the parent function:

2xy

42 xy

Describe the transformation to the parent function:

2xy

52 xy

translated down 4translated down 4

translated up 5translated up 5

Graphical Transformations2xy

CompareCompare the two parabolas. the two parabolas.

MultiplyingMultiplying the parent the parent function by 3, makes it 3 function by 3, makes it 3 times as steep.times as steep.

23xy

2xy

Why does multiplying the parent function by 3 cause the parent to be vertically stretched by a factor of 3?.

23xy

Build a table of values for each equation for domain elements: -2, -1, 0, 1, 2.

x y

-2

-1

0

1

2

x y

-2

-1

0

1

2

41014

12303

12

Graphical Transformations

2xy

CompareCompare the two parabolas. the two parabolas.

MultiplyingMultiplying the parent the parent function by -1, reflects function by -1, reflects across the x-axis.across the x-axis.

2xy

Your Turn:

Describe the transformation to the parent function:

2xy

22 xy

Describe the transformation to the parent function:

2xy

63 2 xy

Reflected across x-axis Reflected across x-axis and translated up 2and translated up 2

Vertically stretched by Vertically stretched by a factor of 3 and a factor of 3 and

translated down 6translated down 6

Graphical Transformations

2xy

CompareCompare the two parabolas. the two parabolas.

2)1( xy

2xy Why does replacing ‘x’ with ‘x – 1’ translates the parent function right by 1.

Build a table of values for each equation for domain elements: -2, -1, 0, 1, 2.

x y-2-1012

x y-2-1012

4

101

4

94101

2)1( xy

Quadratic Transformations

Reflection Reflection across x-axisacross x-axis

translating translating upup or or downdown vertical vertical

stretch stretch factorfactor

khxay 2)()1(

Translates Translates left/rightleft/right

4)3(2 2 xyReflected across x-axis, twice as steep,

translated up 4, translated right 3

Your Turn:Describe the transformation to the parent function:

2xy

3)5( 2 xy

translated up 3 translated up 3 translated left 5translated left 5

Your Turn:Describe the transformation to the parent function:

2xy 2)1(2 xy

Vertically stretched by a factor of 2, Vertically stretched by a factor of 2, translated right 1translated right 1

Your Turn:Describe the transformation to the parent function:

2xy

4)3(21 2 xy

Reflected across x-axis Reflected across x-axis Vertically stretched by a factor of ½ (shrunk), Vertically stretched by a factor of ½ (shrunk),

translated up 4 translated up 4 translated left 3translated left 3

Absolute Value Function xxf )(

Why does it have this shape?

Your turn:

3 xy

xy 2

What is the transformation to the parent function?What is the transformation to the parent function?

xy

Vertically stretched Vertically stretched by a factor of 2 by a factor of 2

Twice as Twice as steepsteep

Slope on right side is +2 slope on left side is -2

translated right 3translated right 3

Your turn:

423 xyWhat is the transformation to the parent function?

xy

Reflected across x-axisReflected across x-axis

VSF = 4 VSF = 4 4 times as steep 4 times as steepLeft 2Left 2

up 4up 4

Reflection Reflection across x-axisacross x-axis

translating translating upup or or downdown

Vertical Vertical stretch stretch factorfactor

Translates Translates left/rightleft/right

khxay )1(Absolute Value Transformation

What does adding or subtraction “k” do to the parent What does adding or subtraction “k” do to the parent function?function? kxxf )(

hxxf )(

What does adding or subtraction “h” do to the parent What does adding or subtraction “h” do to the parent function?function?

What does multiplying by ‘a’ do to the parent What does multiplying by ‘a’ do to the parent function?function?

xaxf )(

Vertical shiftVertical shift

Horizontal shiftHorizontal shift

Vertical stretchVertical stretch

What does multiplying by (-1) do to the parent What does multiplying by (-1) do to the parent function?function?

xxf )( Reflection across x-axisReflection across x-axis

Square Root Function xxf )(

What is the domain of the graph?

Describe the transformation to the parent function:

24 xy

khxay )1(

323 xy

Up 4, right 2Up 4, right 2 42 xy

Down 3, Down 3, reflected across x-axis, reflected across x-axis, VSF=2 VSF=2 left 3 left 3