PreCalculus Class Notes CT4 Transformations of Graphs
Fathom: Polynomial Transformations exploration
Vertical and Horizontal Shifts
Original Graph
Vertical shift up
Horizontal shift right
Horizontal shift left
Example
Shift the graph of y = |x| to the right 2 units and then downward 4 units. Write the new equation.
y = |x|
y=
y=
Example
Find an equation that shifts the graph of f ( x ) =
units.
1 2
x − 4 x + 6 to the left 8 units and upwards 4
2
Reflection of Graphs Across the x- and y-axes
Example
For the representation of f, graph the reflection (1) across
the x-axis and (2) across the y-axis. The graph of f is the
graph determined by the table.
across the x-axis
across the y-axis
Vertical Stretching and Shrinking
y= x
y=
y=2 x
1
x
2
Horizontal Stretching and Shrinking
f ( x ) = x2 − 4 x
x 0 2 4
y 0 4 0
2
⎛1 ⎞
⎛1 ⎞
f ( x) = ⎜ x ⎟ − 4⎜ x ⎟
⎝2 ⎠
⎝2 ⎠
f ( x) = (2x) − 4 (2x)
2
x
y
x
y
Example: Sketch the graph of each equation and rewrite the table.
y = f ( x)
Transformation
(a > 0)
y = 3 f ( x)
Description
⎛1 ⎞
y = f ⎜ x⎟
⎝2 ⎠
Example Graph and Equation
y = f ( x + a)
y = f ( x − a)
Horizontal translation a units
( x + a ) translates graph left
y
y = f(x + a)
y = f(x – a)
( x − a ) translates graph right
y = f(x)
x
y = f ( x) + a
y = f ( x) − a
y
Vertical translation a units
+ a translates graph up
– a translates graph down
y = f(x) + a
y = f(x)
y = f(x) – a
x
y = f (−x)
Reflection over the y-axis
y
x
y = f(–x)
y = − f ( x)
Reflection over the x-axis
y = f(x) y
y = f(x)
x
y = –f(x)
y = af ( x )
y
Vertical dilation by a factor of
a
y = af(x)
y = f(x)
x
⎛x⎞
y= f ⎜ ⎟
⎝a⎠
Horizontal dilation by a factor
of a
y y = f(x) ⎛x⎞
y= f⎜ ⎟
⎝a⎠
x