September 20, 2013
Section 1.6
Graphical Transformations
A transformation changes a graph in a
predictable way. A rigid transformation
leaves the size and shape of the graph
unchanged. Vertical shifts, horizontal shifts
and reflections are rigid transformations.
Vertical and horizontal stretches or shrinks
are non-rigid transformations. These will
distort the shape of the graph.
y=x2
Each graph below is a transformation of
the graph of f(x) = x2. Find is the equation
for each graph.
y=(x+7)2
y=x2+3
y=x2-5
x = -7
y=-x2
y=0
y=(x-4)2
y=(x-1)2-2
September 20, 2013
Vertical Translation c > 0
y = f (x) + c Shift the graph up c units.
y = f (x) − c Shift the graph down c units.
Horizontal Translation c > 0
y = f (x + c) Shift the graph left c units.
y = f (x − c) Shift the graph right c units.
Reflection
y = -f (x)
Reflect across the x-axis.
y = f (-x)
Reflect across the y-axis.
Vertical Stretch or Shrink
y = c ⋅ f (x) Stretch by a factor of c if c > 1.
Shrink by a factor of c if 0 < c < 1.
September 20, 2013
Follow order of operations when combining
transformations. A horizontal shift is
independent of vertical transformations and
can be applied at any time throughout the
process.
y = f (x)
(3, 2)
(-5, 2)
(-5, 0)
Graph each
y = f (x) + 2
y = f (x − 3)
y = f (x − 3) + 2
(3, 0)
(0, -2)
(3, -2)
(0, -4)
(3, -4)
If (2, 5) is on the graph of y = f (x), find a point
on the graph of y = -2 f (x + 4) − 11.
Shift the graph left 4 units. - independent
Stretch the graph by a factor of 2.
Reflect the graph across the x-axis.
- interchangeable
Shift the graph down 11 units. - after stretch & reflection
September 20, 2013
Match the graph with the correct transformations.
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y = f(x)
y = -f(x)
y = f(x) + 2
y = f(x - 1)
y = 2f(x - 1) + 2