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Function Dilations (and Translations)

June 24, 2010

This posting assumes you have read Function Translations.

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, a paper which is also stapled onto a solid surface along one axis. Now grasp the elastic paper with both hands, one hand on each side of the stapled 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 is useful to be able to interpret a function as a dilation of a simpler “parent function”. Understanding the behavior of

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

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

f(x)=x^2

(the solid curve below) dilated vertically by a factor of 3, then translated vertically by +5 and horizontally by +1.

Vert. Dilation with both Vert. and Horiz. Translation

The blue point at the vertex of the solid curve had its y-coordinate (0) multiplied by three (0 x 3 = 0), then five added to it (0 + 5 = 5). It was then shifted one unit to the right, and that point (1,5) remains the vertex of the new curve.

The green point on the solid curve also had its y-coordinate (4) multiplied by three (4 x 3 = 12), then five added to it (12 + 5 = 17). It was then shifted one unit to the right, and that point (3,17) is one which satisfies the equation of  the dashed curve.

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

Vertical Dilation

Consider the graph of a function

x^2

(the solid curve below) translated vertically by +4, so its vertex occurs at the point (0,4) instead of (0,0). Let’s make this our parent function:

f(x)=x^2+4

(the dashed curve below). What happens to the dashed line f(x) if its equation is changed by multiplying all terms by three:

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

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

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

which shows that every point on the graph of f(x), as well as the amount of the parent function’s vertical translation, has been stretched vertically until it is three times farther away from the x-axis. This is shown as a 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(x) have for a value at this same input value of 2?” The original f(x) will be stretched vertically by a factor of three vertically everywhere, including at x=2, so a point on the original graph such as (2,8) becomes (2,24). You can verify for yourself that (2,24) does indeed satisfy the above equation for g(x).

This process works for any function.

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

af(x)

represents a vertical dilation of the parent function by a factor of “a”. Now let’s apply this idea to the following:

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

so

-5sin(x)

represents a vertical dilation by -5 of

sin(x)

Or applying it to a different-looking situation:

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

so

\frac{k}{x}

represents a vertical dilation by k of

\frac{1}{x}

And applying it to an even more complex-appearing situation:

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

so

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

represents a vertical dilation by +3, and a horizontal translation by +1 of

x^2+x+1

Note that the original vertical translation of +1 is affected by the vertical dilation, and becomes +3.

And finally:

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

so

2sin(x-7)+3

represents a vertical dilation by +2, and a horizontal translation by +7, then a vertical translation by +3 (after the vertical dilation is complete) of

sin(x)

Horizontal Dilation

Once again,consider the graph of:

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 we can once again write this second function in terms of the first very compactly, however this time the factor is multiplied by the function’s parameter (instead of the function result, as with vertical dilations):

g(x)=f(3x)

f(x) passed through the point (3,13). Since we are examining horizontal dilations, let’s ask “What must g(x) have for an input to produce this same output?”

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

so (3,13) on the graph of f(x) moves to (1,13) when “3x” is substituted for “x”. You can verify for yourself that the point (1,13) does indeed satisfy the equation

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

But why has the graph been compressed towards the y-axis when the coefficient of “x” was a +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”.

Now 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)

Applying the approach to a quadratic function looks a bit more complex:

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

so

\frac{1}{kx}

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

\frac{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.

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

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