This posting assumes you have read Function Transformations: Translation. Translations represent some combination of vertical and/or horizontal shifts of a function’s graph, but these are only two of the possible transformations of a function.

Vertical Dilation (no Translation)

A function has been “dilated” (note the spelling… it is not spelled or pronounced “dialated”) when it has been stretched away from an axis, or compressed toward an axis. Imagine a graph that has been drawn on elastic paper attached a solid surface along one axis. Now grasp the elastic paper with both hands, one hand on each side of the attached axis, and pull both sides of the graph away from the axis… you are “dilating” the graph, and causing all points to move away from the axis to a multiple of their original distance from the axis.

The graph to the right shows what a function might look like both before and after a vertical dilation. Note that the points on the dashed line are each exactly twice as far from the “x” axis as the same color point on the solid line. Therefore, there are two ways of describing the relationship between the two graphs. Either:

• The solid line has been “dilated vertically by a factor of 2″ to produce the dashed line, or
• The dashed line has been “dilated vertically by a factor of 0.5″ to produce the solid line.

Dilating by a negative factor will cause both a dilation and a reflection about the axis to occur (points that were on one side of the axis are now on the other).

It can be useful to interpret a function as a dilation of a simpler “parent function”. Understanding the behavior of the function

$g(x)=3(x-1)^2+5$

which is graphed as the dashed curve below, is easier when you perceive it as a transformation of a simpler “parent” function:

$f(x)=x^2$

which is graphed as the solid curve below. f(x) has been dilated vertically by a factor of 3, then translated vertically by +5 and horizontally by +1 to produce g(x).

Vert. Dilation with both Vert. and Horiz. Translation

The blue point at the origin, which is the vertex of the solid curve, had its y-coordinate (0) multiplied by three and had five added to it:

(0) x 3 + 5 = 5

It was then shifted one unit to the right, causing its x-coordinate to change from being 0 to 1. So, the old vertex that was at the origin is now the vertex of the new curve and is located at (1, 5).

The green point on the solid curve (2, 4) also had its y-coordinate (4) multiplied by three and had five added to it:

4 x 3 + 5 = 17

It was then shifted one unit to the right, just as the vertex was, and that point (3 ,17) satisfies the equation of the dashed curve, g(x).

Visualizing functions as translations and dilations of a simpler “parent function” can make complex-looking equations much easier to interpret.

Vertical Dilation

Consider the solid graph below, which represents the function:

$x^2$

If it is translated vertically by +4, so that its vertex occurs at the point (0,4) instead of (0,0), the equation becomes:

$f(x)=x^2+4$

which is graphed by the dashed curve below. What happens to the dashed line f(x) if every term in its equation is multiplied by three?

$g(x)=3x^2+3\cdot 4$

Note that we could easily write this second function in terms of the first:

$g(x)=3\cdot f(x)$

By defining g(x) this way, we are demonstrating that every point on the graph of f(x) has been stretched vertically until it is three times farther away from the x-axis, as shown in the uppermost (dotted) curve below.

Vert. Dilation of a Vert. Translated Parent Function

f(x) passes through the point (2,8). Since we are examining vertical dilations, let’s keep the x-coordinate the same and ask “What will g(2) be?” The original f(x) will be stretched vertically by a factor of three vertically everywhere, including at x=2, so (2,8) becomes (2,24). You can verify for yourself that (2,24) satisfies the above equation for g(x).

This process works for any function.

Any time the result of a parent function is multiplied by a value, the parent function is being vertically dilated. If f(x) is the parent function, then

$af(x)$

dilates f(x) vertically by a factor of “a”.

Let’s apply this idea to a trigonometry function:

$f(x)=sin(x)\\*~\\*g(x)=-5\cdot f(x)=-5sin(x)$

Based on the explanation in the previous paragraph, we can conclude that

$-5sin(x)$

represents a vertical dilation by -5 of

$sin(x)$

If we apply this approach to another type function

$f(x)=\dfrac{1}{x}\\*~\\*~\\*g(x)=k\cdot f(x)=\dfrac{k}{x}$

you can see that we can analyze it the same way:

$\dfrac{1}{x}$

dilated vertically by a factor of k becomes:

$\dfrac{k}{x}$

Applying this approach to an even more complex situation:

$f(x)=x^2+x+1\\*~\\*g(x)=3\cdot f(x-1)=3(x-1)^2+3(x-1)+3\cdot 1$

The parent function in this case is

$x^2+x+1$

and every instance of “x” in this function has had (x-1) substituted for it, which translates it horizontally by +1. Then this result was multiplied by 3, causing a vertical dilation by a factor of 3:

$3(x-1)^2+3(x-1)+3$

Note that the original vertical translation of +1 (the constant term in the parent function’s definition) is also affected by the vertical dilation, and becomes +3 in the final equation above.

One last example:

$f(x)=sin(x)\\*~\\*g(x)=2\cdot f(x-7)+3=2sin(x-7)+3$

The parent function

$sin(x)$

has been dilated vertically by a factor of +2, translated horizontally by +7, and then translated vertically by +3 (after being dilated vertically):

$2sin(x-7)+3$

Horizontal Dilation

$f(x)=x^2+4$

What happens to this graph if the equation is changed by multiplying every “x” in the equation by three:

$g(x)=(3x)^2+4$

Note that, once again, we can describe g(x) more compactly if we do so using f(x), however this time the dilation factor is multiplied by the function’s “input variable” instead of its “result” (as was done to produce a vertical dilation):

$g(x)=f(3x)$

Note that f(x) passes through the point (3,13). Since we are thinking about horizontal dilations, let’s ask “What value must ‘x’ have if g(x) is to produce this same result of 13?”

$13=(3x)^2+4\\*~\\*9=9x^2\\*~\\*1=x^2\\*~\\*1=x$

This shows that the point (3,13) on the graph of f(x) corresponds to the point (1,13) on g(x). Verify for yourself that the point (1,13) satisfies the equation for g(x):

$g(x)=(3x)^2+4$

If multiplying the result of a function by a factor causes a vertical dilation by the same factor, why does multiplying the input variable by a factor cause a horizontal dilation by the reciprocal of that factor? The graph in this example has been compressed horizontally, moving to one third of its previous distance from the y-axis when every “x” was multiplied by 3. If using a coefficient greater than one expands things vertically, why does it shrink things horizontally? This difference in effect seems counter-intuitive at first glance.

The difference occurs because vertical dilations occur when we scale the output of a function, whereas horizontal dilations occur when we scale the input of a function.

The “x” in the original equation became a “3x”, so the new equation reaches a given “input value” three times faster. “x” only has to be 1/3 as big in g(x) for the result of the equation to be the same as f(x). Therefore, all points on g(x) have been scaled to be 1/3 of the distance from the vertical axis that they were in f(x).

This process works for any function. Anytime the input of the “parent function” is multiplied by a value, the parent function is being horizontally dilated. If

$f(x)$

is the parent function, then

$f(ax)$

represents a horizontal dilation of the parent function by a factor of “1/a”.

Apply this idea to a slightly more complex situation:

$f(x)=sin(x)\\*~\\*g(x)=f(5x)\\*~\\*g(x)=sin(5x)$

so

$sin(5x)$

represents a horizontal dilation by a factor of 1/5 (toward the vertical axis) of

$sin(x)$

In other words, the period of f(x) was $2\pi$, and the period of g(x) is $\dfrac{2\pi}{5}$

Horizontal dilations of a quadratic function look a bit more complex at first, until you become accustomed to the pattern you are looking for:

$f(x)=x^2-x\\*~\\*g(x)=f(\frac{1}{2}x)\\*~\\*g(x)=(\frac{1}{2}x)^2-(\frac{1}{2}x)$

so

$(\frac{1}{2}x)^2-(\frac{1}{2}x)$

represents a horizontal dilation by a factor of 2 (away from the vertical axis) of

$x^2-x$

Note that every instance of “x” in the parent function must be changed to be

$\frac{1}{2}x$

for the new equation to represent a horizontal dilation of the parent by a factor of 2.

Applying this approach to a fractional situation:

$f(x)=\dfrac{1}{x}\\*~\\*~\\*g(x)=f(kx)\\*~\\*~\\*g(x)=\dfrac{1}{kx}$

so

$\dfrac{1}{kx}$

represents a horizontal dilation by a factor of 1/k of

$\dfrac{1}{x}$

What’s The Difference?

In contemplating both vertical and horizontal dilations, you may have realized that the graphs of some functions, such as

$y=(2x)^2=4x^2$

could be considered either a vertical dilation by a factor of 4 or a horizontal dilation by a factor of 1/2. It is interesting to note that both dilations, stretching it vertically or squeezing it horizontally, have the same end result for this function. Can this be true for other functions as well? Consider the following equivalent equations:

$y=(6x-12)^2\\*~\\*=(6[x-2])^2~~~see~(1)~below\\*~\\*=(2\cdot 3[x-2])^2\\*~\\*=4(3[x-2])^2~~~see~(2)~below\\*~\\*=4\cdot 9[x-2]^2\\*~\\*=36[x-2]^2~~~see~(3)~below$

This example demonstrates that some functions can transformed to the same end result by either a horizontal dilation, a vertical dilation, or a combination of both. In the example above, the following three sets of dilations and translations of the parent function $y=x^2$ produce the same graph:
1)  Dilated horizontally by a factor of 1/6, then translated horizontally by +2. No vertical dilation.
2) Dilated horizontally by a factor of 1/3, then translated horizontally by +2. Dilated vertically by a factor of 4.
3)  No horizontal dilation, translated horizontally by +2. Dilated vertically by a factor of 36.

Note how the horizontal translations change as the horizontal dilations change. Since a horizontal dilation shrinks the entire graph towards the vertical axis, the graph’s horizontal translation shrinks by the same factor. As the original horizontal dilation factor of 1/6 in the example above is increased by a factor of 6 to be 1 (becoming converted into a vertical dilation factor of 36 in the process), the original horizontal translation of 12 shrinks by a factor of 6 to become 2.

So which of all the above options is the “normal” way of describing this graph? Having a preferred way of describing it will make it more likely that different people will describe the graph in the same way…

The “normal” way of describing a combination of dilations and translations is to convert all dilations into vertical dilations by manipulating the expression so that the independent variable has a coefficient of one:

$y=(6x-12)^2\\*~\\*y=(6[x-2])^2\\*~\\*y=36[x-2]^2$

So this equation represents a vertical dilation by a factor of 36 and a horizontal translation of +2 of the equation

$y=x^2$

If you were not interested in the vertical dilation, but only in the horizontal translation, you could solve the independent variable expression (before applying any exponent) for zero:

$y=(6x-12)^2\\*~\\*0=6x-12\\*~\\*12=6x\\*~\\*2=x$

which tells us that the “parent function” has been translated horizontally by +2 after all dilations have been carried out.

Dilation About Lines Away From An Axis

In some situations it will be useful to dilate a function relative to a horizontal or vertical line other than the axis.  To achieve this, we need to:

1. Translate the graph so that the part of the graph that is to remain unchanged by the dilation is moved to the axis
2. Dilate the graph by the desire amount
3. Translate the dilated function back to its original location

Suppose we wish to dilate a function f(x) vertically by a factor of 3 about the line y=2. The above steps produce the following for the function f(x):

Vertical dilation about the line y=2 by a factor of 3

$f(x)=x^2$

Translate f(x) down 2, so that the line about which we wish to dilate is moved onto the x-axis:

$f(x)-2$

Dilate the translated function vertically by a factor of 3:

$3(f(x)-2)$

Now “undo” the original vertical translation by translating it back up 2:

$3(f(x)-2)+2\\*~\\*g(x)=3f(x)-4$

If you graph both f(x) and g(x) on the same graph, as shown to the right, you will note that the two graphs intersect one another at the line y=2, which is the line about which we dilated f(x). Those are the only two points on the graph of f(x) that remain unchanged by the dilation.

This same process can be followed to create horizontal dilations about some vertical line: translate the function horizontally, then dilate it, then translate the result back to where it started.

Want to Play?

If you would like to play around with vertical dilations and see how they work, try any of the following Geogebra applets.  The only one that lets you play with horizontal dilations is the last one (Sine Function):
Quadratic function in vertex form
Exponential function
Sine function