Transforming Graphs of Functions
Note: Throughout this section we shall take f(x) = x2 – 2x – 3
This is a parabola that cuts the X-axis at (-1,0) and (3,0)
and has a min TP at (1,-4) as shown in the diagram.
y = x2
– 2x – 3
Comparing f(x) with f(x) + k
Taking
f(x) = x2 – 2x – 3
then y = f(x) + 5
is y = (x2 – 2x – 3) + 5
ie y = x2 – 2x + 2
Key points on this graph are (-1,5), (3,5) & (1,1).
The original graph has moved up 5 spaces.
y = x2 – 2x + 2
Comparing f(x) with f(x) – k
Taking
f(x) = x2 – 2x – 3
then y = f(x) - 3
is y = (x2 – 2x – 3) - 3
ie y = x2 – 2x - 6
Key points on this graph are (-1,-3), (3,-3) & (1,-7).
The original graph has moved down 3 spaces.
y = x2 – 2x - 6
SUMMARY
To obtain the graph of y = f(x) + k we move the graph
of y = f(x) k units up.
To obtain the graph of y = f(x) - k we move the graph
of y = f(x) k units down.
Comparing f(x) with f(x+k)
Taking
f(x) = x2 – 2x – 3
then y = f(x+3)
is y = (x+3)2 – 2(x+3) – 3
is y = x2 + 6x + 9 - 2x - 6 – 3
ie y = x2 + 4x
Key points on this graph are (-4,0), (0,0) & (-2,-4).
The original graph has moved 3 spaces to the left.
y = x2 + 4x
Comparing f(x) with f(x-k)
Taking
f(x) = x2 – 2x – 3
then y = f(x-2)
is y = (x - 2)2 – 2(x - 2) – 3
is y = x2 - 4x + 4 – 2x + 4 – 3
ie y = x2 – 6x + 5
Key points on this graph are (1,0), (5,0) & (3,-4).
The original graph has moved 2 spaces to the right.
y = x2 – 6x + 5
SUMMARY
To obtain the graph of y = f(x + k) we move the graph
of y = f(x) k units to the left.
To obtain the graph of y = f(x – k) we move the graph
of y = f(x) k units to the right.
Comparing f(x) with -f(x)
Taking
f(x) = x2 – 2x – 3
then y = -f(x)
is y = -(x2 – 2x – 3)
ie y = -x2 + 2x + 3
Key points on this graph are (-1,0), (3,0) & (1,4).
The original graph has been reflected in the X-axis.
y = -x2 + 2x + 3
Comparing f(x) with f(-x)
Taking
f(x) = x2 – 2x – 3
then y = f(-x)
is y = (-x)2 – 2(-x) – 3)
ie y = x2 + 2x - 3
Key points on this graph are (-3,0), (1,0) & (-1,-4).
The original graph has been reflected in the Y-axis.
y = x2 + 2x - 3
SUMMARY
To obtain the graph of y = -f(x) we reflect the graph
of y = f(x) in the X-axis.
To obtain the graph of y = f(-x) we reflect the graph
of y = f(x) in the Y-axis.
Comparing f(x) with kf(x)
Taking
f(x) = x2 – 2x – 3
then y = 2f(x)
is y = 2(x2 – 2x – 3)
ie y = 2x2 – 4x - 6
Key points on this graph are (-1,0), (3,0) & (1,-8).
The original graph has moved up “stretched” vertically
by a factor of 2. (ie it is twice as tall!)
y = 2x2 – 4x - 6
Taking
f(x) = x2 – 2x – 3
then y = 0.5f(x)
is y = 0.5(x2 – 2x – 3)
ie y = 0.5x2 – x - 1.5
Key points on this graph are (-1,0), (3,0) & (1,-2).
The original graph has moved up “squashed” vertically
by a factor of 2. (ie it is half as tall!)
y = 0.5x2 – x - 1.5
SUMMARY
To obtain the graph of y = kf(x) (where k > 1) we “stretch”
the graph of y = f(x) vertically by a factor of k.
To obtain the graph of y = kf(x) (where 0 < k < 1) we “squash”
the graph of y = f(x) vertically by a factor of k.
Comparing f(x) with f(kx)
Taking
f(x) = x2 – 2x – 3
then y = f(2x)
is y = (2x)2 – 2(2x) – 3
ie y = 4x2 – 4x - 3
Key points on this graph are (-0.5,0), (1.5,0) & (0.5,-4).
The original graph has been “squashed” horizontally
by a factor of 2. (ie it is half as wide!)
y = 4x2 – 4x - 3
Taking
f(x) = x2 – 2x – 3
then y = f(0.5x)
is y = (0.5x)2 – 2(0.5x) – 3
ie y = 0.25x2 – x - 3
Key points on this graph are (-2,0), (6,0) & (2,-4).
The original graph has been “stretched” horizontally
by a factor of 2. (ie it is twice as wide!)
y = 0.25x2 – x - 3
SUMMARY
To obtain the graph of y = f(kx) (where k > 1) we “squash”
the graph of y = f(x) horizontally by a factor of k.
To obtain the graph of y = f(kx) (where 0 < k < 1) we “stretch”
the graph of y = f(x) horizontally by a factor of k.