Section 3.5 Transformations
Vertical Shifts (up or down)
y  f x   k
Graph, given f(x) = x2.
y  f x   2
y  x2  2
Every point shifts
up 2 squares.
y  f x   4
y  x2  4
Every point shifts
down 4 squares.
Section 3.5 Transformations
Horizontal Shifts (left or right)
y  f x  h 
Graph, given f(x) = x2.
y  f  x  2
Every point shifts
2
left 2 squares.
y  x  2
y  f  x  4
Every point shifts
2
right 4 squares.
y  x  4
Section 3.5 Transformations
Vertical Stretching and shrinking.
y  a  f x 
(-2, 8)
(2, 8)
Graph, given f(x) = x2.
y  2 f x 
y  2x 2
Every y coordinate
of each point is
multiplied by 2.
1
y  f x 
2
Every y coordinate
1 2 of each point is
y  x multiplied by 1/2.
2
(-2, 4)
(2, 4)
(-1, 2)
(-2, 2)
(-1, 1)
(-1, 1/2)
(1,(2,
2) 2)
(1, 1)
(0, 0)
(1, 1/2)
Section 3.5 Transformations
Horizontal Stretching and shrinking.
y  f b  x
Everything is opposite of vertical!
Graph, given f(x) = |x|.
y  f 2 x 
y  2x
Every x coordinate
(-2, 4)
of each point is
(-4, 4)
(-3, 3) (-1, 2)
divided by 2.
(-6, 3)
(-4, 2) (-2, 2)
1 
y  f  x
 2  Every x coordinate
1
y x
2
(2, 4)
(4, 4)
(1, 2) (3, 3)
(6, 3)
(2, 2) (4, 2)
(0, 0)
of each point is
divided by 1/2.
Remember we don’t divide by fractions,
we multiply by the reciprocal!
Section 3.5 Transformations
Reflections, flipping over x-axis or y-axis.
Graph, given f x   x.
y   f x 
Every point will
flip over the x-axis.
y x
y  f  x 
y  x
Every point will
flip over the y-axis.
Transformations have a specific order…
The ORDER OF OPERATIONS!
3rd
1st
2nd
4th
Outside the function…
Inside the function…
affects y-coordinates.
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
EXAMPLE.
5.
y  2 ( x  5) 3
1. Flips over y-axis.
1.
2.
2. – x + 5 = 0
+5=x
Right 5 units.
3. Flips over x-axis.
4. Vertical Stretch by 2.
5. Up 3 units.
Consider the function y  f x  on the graph.
Graph y  f x  2  3 .
1. x – 2 = 0.
x = +2
Right 2 units.
2. Up 3 units
Graph y  f  x   5 .
Consider the function y  f x  on the graph.
Graph y  f x  2  3 .
1. x – 2 = 0.
x = +2
Right 2 units.
2. Up 3 units
Graph y  f  x   5 .
1. negative on the inside flips
over the y-axis.
2. – 5 on the outside shifts
down 5 units
Consider the function y  f x  on the graph.
Graph y  3 f x  .
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. A quicker way is to multiply all
y-coordinates by -3.
(1, 3)
(-3, 2)
(3, 3)
(-1, 1)
1 

Graph y  f  x  .
2 
(1, -1)
(3, -1)
(-1, -3)
(-3, -6)
Consider the function y  f x  on the graph.
Graph y  3 f x  .
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. A quicker way is to multiply all
y-coordinates by -3.
1 

Graph y  f  x  .
2 
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).
(1, 3)
(-6, 2)
(-3, 2)
(3, 3)
(-1, 1)
(-2, 1)
(1, -1)
(-1, -3)
(-3, -6)
(3, -1)
(2, -1)
(6, -1)
Consider the function y  f x  on the graph.
Graph y  f  2 x  .
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)
Graph y  2 f x  3 .
(-1, 1)
(-1.5, -1)
(1.5, 2)
(0.5, 1)
(1, -1)
(-0.5, -1)
(3, -1)
Consider the function y  f x  on the graph.
Graph y  f  2 x  .
(-6, 4)
(-4, 2)
(-3, 2)
(-1, 1)
(-6, 2)
Graph y  2 f x  3 .
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.
(-4, 1)
(-2, -1)
( 0, -1)
(1, -1)
(3, -1)
(-2, -2) ( 0, -2)