Section 3.5 Transformations Vertical Shifts (up or down) Graph, given f(x) = x 2. Every point shifts...
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Transcript of Section 3.5 Transformations Vertical Shifts (up or down) Graph, given f(x) = x 2. Every point shifts...
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)