Section 3.5 Transformations

Vertical Shifts (up or down)

kxfy

2 xfy

4 xfy

Graph, given f(x) = x2.

22 xy

42 xy

Every point shifts up 2 squares.

Every point shifts down 4 squares.

Section 3.5 Transformations

Horizontal Shifts (left or right)

hxfy

2 xfy

4 xfy

Graph, given f(x) = x2.

22 xy

24 xy

Every point shifts left 2 squares.

Every point shifts right 4 squares.

Section 3.5 Transformations

Vertical Stretching and shrinking.

xfay

xfy 2

xfy2

1

Graph, given f(x) = x2.

22xy

2

2

1xy

Every y coordinate of each point is multiplied by 2.

Every y coordinate of each point is multiplied by 1/2.

(1, 2)(-1, 2)

(-2, 8) (2, 8)

(0, 0)

(1, 1)(-1, 1)

(-2, 4) (2, 4)

(1, 1/2)(-1, 1/2)

(-2, 2) (2, 2)

(0, 0)(0, 0)

Section 3.5 Transformations

Horizontal Stretching and shrinking.

xbfy

xfy 2

xfy

2

1

Graph, given f(x) = |x|.

xy 2

xy2

1

Every x coordinate of each point is divided by 2.

Every x coordinate of each point is divided by 1/2.

(1, 2)(-1, 2)

(-2, 4) (2, 4)

(0, 0)

(3, 3)(-3, 3)

(-2, 2) (2, 2)(6, 3)(-6, 3)

(-4, 2) (4, 2)

(0, 0)(0, 0)

Everything is opposite of vertical!

Remember we don’t divide by fractions, we multiply by the reciprocal!

(4, 4)(-4, 4)

Section 3.5 Transformations

Reflections, flipping over x-axis or y-axis.

xfy Graph, given .

xy

Every point will flip over the x-axis.

Every point will flip over the y-axis.

xxf

xfy

xy

Transformations have a specific order…The ORDER OF OPERATIONS!

1st 2nd3rd 4th

Outside the function… affects y-coordinates.

1. If a is negative, then flips over x-axis.

2. If | a | is > 1, then Vertical Stretch. If 0 < | a | < 1, then Vertical Shrink.

Inside the function… affects x-coordinates.

1. If b is negative, then flips over y-axis.

2. If | b | is > 1, then Horizontal Shrink. If 0 < | b | < 1, then Horizontal Stretch.

Inside the function… affects x-coord.

Solve bx – c = 0. The answer for x will tell you which direction (sign) and how far (value).

Outside the function… affects y-coord.

Take the value of d for face value.+ d goes up d units; – d goes down d units.

EXAMPLE. 352 xy1. Flips over y-axis.

1.

2. – x + 5 = 0 + 5 = xRight 5 units.

( )2.

3. Flips over x-axis.

3 & 4

4. Vertical Stretch by 2.

5.

5. Up 3 units.

Consider the function on the graph.

1. x – 2 = 0. x = +2Right 2 units.

xfy

Graph . 32 xfy2. Up 3 units

Graph . 5 xfy

Consider the function on the graph.

1. x – 2 = 0. x = +2Right 2 units.

1. negative on the inside flips over the y-axis.

xfy

Graph . 32 xfy2. Up 3 units

Graph . 5 xfy

2. – 5 on the outside shifts down 5 units

Consider the function on the graph.

1. Negative on the 3 will flip the graph over the x-axis and | -3 | = 3 and will cause a vertical stretch by multiplying the y-coordinates by 3.

xfy

Graph . xfy 3

Graph .

xfy

2

1

1. A quicker way is to multiply all y-coordinates by -3.

(3, -1)

(1, -1)

(-1, 1)(-3, 2)

(-3, -6)

(-1, -3)

(1, 3)(3, 3)

Consider the function on the graph.

1. Negative on the 3 will flip the graph over the x-axis and | -3 | = 3 and will cause a vertical stretch by multiplying the y-coordinates by 3.

xfy

Graph . xfy 3

Graph .

xfy

2

1

1. A quicker way is to multiply all y-coordinates by -3.

(3, -1)

(1, -1)

(-1, 1)(-3, 2)

(-3, -6)

(-1, -3)

(1, 3)(3, 3)

1. Multiplying ½ to the inside will cause a horizontal stretch. Remember, everything is opposite.Divide all x-coordinates by ½. Again we don’t divide by fractions, instead we multiply by the reciprocal ( times by 2).

(-6, 2)

(-2, 1)

(2, -1) (6, -1)

Consider the function on the graph.

1. Negative on the 2 will flip the graph over the y-axis and | -2 | = 2 and will cause a horizontal shrink by dividing the x-coordinates by 2.

xfy

Graph . xfy 2

Graph . 32 xfy

1. A quicker way is to divide all x-coordinates by -2.

(3, -1)

(1, -1)(-1, 1)

(-3, 2) (1.5, 2)(0.5, 1)

(-0.5, -1)

(-1.5, -1)

Consider the function on the graph. xfy

Graph . xfy 2

Graph . 32 xfy(3, -1)

(1, -1)

(-1, 1)(-3, 2)

(-6, 2)

(-4, 1)

(-2, -1)( 0, -1)

1. The “+3” on the inside of the ( )’s will move every x-coordinate to the left 3 units.

2. The multiplication of 2 on the outside will multiply 2 to every y-coordinate.

(-6, 4)

( 0, -2)(-2, -2)

(-4, 2)