1. Section 3.5 Transformations
Vertical Shifts (up or down)
Graph, given f(x) = x2.
Every point shifts up 2 squares.
Every point shifts down 4 squares.

2. Section 3.5 Transformations
Horizontal Shifts (left or right)
Graph, given f(x) = x2.
Every point shifts left 2 squares.
Every point shifts right 4 squares.

3. Section 3.5 Transformations
Vertical Stretching and shrinking.
(-2, 8)
(2, 8)
Graph, given f(x) = x2.
Every y coordinate of each point is multiplied by 2.
(-2, 4)
(2, 4)
(-1, 2)
(1, 2)
(2, 2)
(-2, 2)
(-1, 1)
(1, 1)
(-1, 1/2)
(1, 1/2)
(0, 0)
(0, 0)
(0, 0)
Every y coordinate of each point is multiplied by 1/2.

4. Section 3.5 Transformations
Horizontal Stretching and shrinking.
Everything is opposite of vertical!
Graph, given f(x) = |x|.
Every x coordinate of each point is divided by 2.
(-2, 4)
(2, 4)
(-4, 4)
(4, 4)
(-3, 3)
(-1, 2)
(1, 2)
(3, 3)
(-6, 3)
(6, 3)
(-4, 2)
(-2, 2)
(2, 2)
(4, 2)
(0, 0)
(0, 0)
(0, 0)
Every x coordinate of each point is divided by 1/2.
Remember we don’t divide by fractions, we multiply by the reciprocal!

5. Section 3.5 Transformations
Reflections, flipping over x-axis or y-axis.
Graph, given
Every point will flip over the x-axis.
Every point will flip over the y-axis.

6. ( ) Transformations have a specific order… The ORDER OF OPERATIONS!
1st
2nd
4th
Outside the function… affects y-coordinates.
Inside the function… affects x-coordinates.
1. If a is negative, then flips over x-axis.
1. If b is negative, then flips over y-axis.
2. If | a | is > 1, then Vertical Stretch.
If 0 < | a | < 1, then Vertical Shrink.
2. If | b | is > 1, then Horizontal Shrink.
If 0 < | b | < 1, then Horizontal Stretch.
Inside the function… affects x-coord.
Outside the function… affects y-coord.
Solve bx – c = 0. The answer for x will tell you which direction (sign) and how far (value).
Take the value of d for face value.
+ d goes up d units; – d goes down d units.
3 & 4 5.
( ) 2. 1.
EXAMPLE.
1. Flips over y-axis.
2. – x + 5 = 0
+ 5 = x
Right 5 units.
3. Flips over x-axis.
5. Up 3 units.
4. Vertical Stretch by 2.

7. Consider the function on the graph.
1. x – 2 = 0.
x = +2
Right 2 units.
2. Up 3 units
Graph

8. Consider the function on the graph.
1. x – 2 = 0.
x = +2
Right 2 units.
2. Up 3 units
Graph
1. negative on the inside flips over the y-axis.
2. – 5 on the outside shifts down 5 units

9. 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.
(1, 3)
1. A quicker way is to multiply all y-coordinates by -3.
(3, 3)
(-1, 1)
(-3, 2)
Graph
(1, -1)
(3, -1)
(-1, -3)
(-3, -6)

10. 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.
(1, 3)
(3, 3)
1. A quicker way is to multiply all y-coordinates by -3.
(-3, 2)
(-6, 2)
(-1, 1)
(-2, 1)
Graph
(1, -1)
(3, -1)
(6, -1)
(2, -1)
(-1, -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).
(-3, -6)

11. 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.
1. A quicker way is to divide all x-coordinates by -2.
(-3, 2)
(1.5, 2)
(-1, 1)
(0.5, 1)
Graph
(1, -1)
(-1.5, -1)
(3, -1)
(-0.5, -1)

12. Consider the function on the graph.
(-6, 4)
(-4, 2)
(-3, 2)
(-6, 2)
(-1, 1)
(-4, 1)
Graph
(1, -1)
( 0, -1)
(-2, -1)
(3, -1)
1. The “+3” on the inside of the ( )’s will move every x-coordinate to the left 3 units.
(-2, -2)
( 0, -2)
2. The multiplication of 2 on the outside will multiply 2 to every y-coordinate.