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“How to” sheet: Curve Transformations
Try to understand the logic behind transformations rather than just remembering a set of
rules.



All your curve transformations either translate or stretch the curve
A transformation involving multiplication must be a stretch (zero values are still zero, don’t
move!)
A transformation involving adding or subtracting must therefore be a translation.
Imagine that you know one pair of coordinates on the curve y  f  x  , perhaps (2,7).
This defines f  2   7 .
Now you are asked to do a transformation. Think: does the transformation change the input to f
or the output from f?
 Transformations that first find f  x  , then add a constant or multiply by a constant are giving
you a new y-value at the same x as before (translation or stretch in the y-direction, points moving
along a vertical, constant x line).
Input value x
f
+7
Output value f  x   7
Input value x
f
×5
Output value 5  f  x 
 Transformations that first modify the x-value and then pass this new number into the function
need a different x-value to give you the same y-value as before (translation or stretch in the xdirection, points move along a horizontal, constant y line)
Input value x
-2
Input value x
×2
x2
2x
f
Output value f  x  2 
f
Output value f  2 x 
Translation
f  x  a
Stretch
y-direction
x-direction
f  x  a
f  kx 
kf  x 
Translation in the y-direction (along a line of constant x)
Defining f  x   x (a very simple function!) and a point on it with known coordinates, (1, 1). We get a
family of lines that are shifted up or down from the initial line. For each line, I will show it both as a
transformation of f(x) and as a line equation (just to help you remember it).
y
y  x4
y  f  x  4
(1,7)
y  x6
6
y  f  x  6
y  x2
y  f  x  2
(1,5)
4
yx
(1,3)
y  f  x
2
Up
−4
(1,1)
x
y  x 43
2
y  f  x  3
−2
−2
(1, 2)
x 1
−4
Stretch in the y-direction (multiplying all the original y-values by a constant)
Plotting y  kf  x  we get a family of lines with y-values that are multiples of the original y-value. All
coordinates are stretched in the y-direction (away from the x-axis) by a factor k.
yy
6
4
 6 f  x
y  4 f  x
y  4x
y  6x
y  2 f  x
y  2x
(1, 4)
yx
2
(1, 2)
Stretch
(1,1)
−4
x
(1, 1)
−2
2
4
y  x
−2
−4
y  f  x
y   f  x
x 1
Translation in the x-direction
y  f  x  a  is the curve y  f  x  translated by a in the  x direction
E.g. y  f  x  1 , a point moves horizontally  a    1  1 to the right along a line of constant y.
To make this y-value, we must start with a bigger x-value than before (because it then has 1 taken off it).
3
I will take the cubic f  x   x  x as an example.
x  0.25, x  1  1.25
x  2.25, x  1  1.25,
  0.7
1f 1.25
y
f 1.25   0.7
 0.25, 0.7 
1.25, 0.7 
 2.25, 0.7 
0.5
x
−2
y  f  x  1
Translation by -1
−1
1
−0.5
y  f  x
y  f  x  1
−1
Translation by +1
Geogebra applets to show you the curves moving
(www.rwmoss.co.uk/Geogebra/curve_transformations_kx+a.ggb,
www.rwmoss.co.uk/Geogebra/bell_transforms.ggb )
Geogebra is free from www.geogebra.org.
2
Stretch in the x-direction
y  f  kx  is the curve y  f  x  stretched by a factor 1k in the  x direction.
The frequency is increased by the factor k (meaning k times as many wiggles in a given x-distance)
E.g. y  f  2 x  , a point moves horizontally along a line of constant y. To make this y-value, we must
start with an x-value that is half of the original (because it is then doubled).
3
I will take the cubic f  x   x  x as an example.
x  0.625, 2x  1.25
f 1.25   0.7
x  2.5,
y
 0.625, 0.7 
0.5
1.25, 0.7 
 12
1
2
f 1.25   0.7
 2.5, 0.7 
2
x
−2
y f
−1
 12 x 
1
2
−0.5
y  f  x
y  f  2x
−1
Always check that your x-coordinate, passed through the line definition, gives you the y-value
you have chosen (e.g. the x  0.625, 2 x  1.25, f 1.25   0.7 boxes above).
x  1.25