Function Dilations: How to recognize and analyze them

For the approach I now prefer to this topic, which uses transformation equations, please follow this link: Function Transformations: Dilation

This post explores one type of function transformation: “dilation”. If you are not familiar with “translation”, which is a simpler type of transformation, you may wish to read Function Translations: How to recognize and analyze them first.

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 graph paper, and fastened to a solid surface along one of the axes. Now grasp the elastic paper with both hands, one hand on each side of the axis that is fixed to the surface, and pull both sides of the paper away from the axis. Doing so “dilates” the graph, causing all points to move away from the axis to a multiple of their original distance from the axis. As an example of this, consider the following graph:

The graph above shows a function before and after a vertical dilation. The coordinates of two points on the solid line are shown, as are the coordinates of the two corresponding points on the dashed line, to help you verify that the dashed line is exactly twice as far from the x-axis as the same color point on the solid line.

The origin is a point shared by both lines, and it is useful to note that the dashed line is still “twice as far from the x-axis” at the origin, because $2 \cdot 0=0$ . Any point that satisfies a function definition and lies on the x-axis will not move when the function is dilated vertically.

There are two ways we can describe the relationship between the two functions graphed above. 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.

Both statements describe the graph accurately. However, in general the function definition which is simplest (in algebraic terms) will be considered the “parent” function, with the more complex-looking definition being described as a dilation of the simpler function.

For example,

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

(graphed as the dashed curve below), is easier to analyze if you perceive it as related to a simpler “parent” function:

$f(x)=x^2$

(graphed as the solid curve below) which has been both dilated and translated:

Vert. Dilation with both Vert. and Horiz. Translation

f(x) has been dilated vertically by a factor of 3, then translated vertically by +5 and horizontally by +1 to produce g(x).

The blue point at the origin, which is the vertex of the solid parabola, had its y-coordinate (0) multiplied by three then 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 0 to 1. So, the “parent” vertex that was at the origin is located at (1, 5) in the transformed function.

The green point on the solid parabola (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 parabola, g(x).

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

Note that a negative dilation factor causes both a dilation and a reflection about the axis to occur. All points that were on one side of the axis of dilation are reflected to the other side of the axis by a negative dilation factor.

Vertical Dilation

Consider the solid parabola below, which represents the function:

$x^2$

If it is translated vertically by +4, so that its vertex moves from (0,0) to (0,4), the equation becomes:

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

which is graphed by the dashed parabola below. What happens to graph of the dashed parabola f(x) if every term in its equation is multiplied by three? We’ll refer to the result of this multiplication as g(x):

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

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 explicitly stating that every y-coordinate produced by g(x) will be three times the corresponding y-coordinate on f(x). In other words, g(x) is f(x) dilated vertically by a factor of three.

Every point on the graph of g(x) below (the upper, dotted, parabola) is three times farther away from the x-axis than the corresponding point on f(x):

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

$a\cdot f(x)$

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

Let’s apply this idea to a trigonometric function:

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

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

$-5\cdot sin(x)$

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:

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

The parent function in this case is

$x^2+x+1$

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

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

The original vertical translation and y-intercept of +1 ( the constant term in the definition of f(x) ) is also affected by the vertical dilation, and becomes +3 in g(x)… three times the distance from the x-axis that it was originally.

One last example:

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

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), to produce g(x):

$2\cdot sin(x-7)+3$

Horizontal Dilation

Let’s return to 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:

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

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 output 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):

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

Since (3,13) moved to (1,13), multiplying every “x” in f(x) by 3 has compressed the graph horizontally, with each point being moved to one third of its previous distance from the y-axis.

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? To ask the question another way, 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 f(x) became a “3x” in g(x), so g(x) reaches a given “input value” three times faster than f(x). “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

$f(a\cdot x)$

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

Apply this idea to a slightly more complex situation:

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

$sin(5x)$

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) is $2\pi$ , 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:

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

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

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:

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

$\dfrac{1}{kx}$

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:

Translate the graph so that the part of the graph that is to remain unchanged by the dilation is moved to the axis
Dilate the graph by the desire amount
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):

$f(x)=x^2$

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:

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

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

If you graph both f(x) and g(x) on the same graph, as shown above, 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

You may also be interested in a topic based on the ideas in this post:
– Using Corresponding Points to Determine Dilation Factors and Translation Amounts

John,

Some texts do not use the word dilation, choosing instead to use phrases such as “Stretching or Shrinking”, or “Stretching or Compressing”, or “Scale Change”.

References to three texts on my bookshelf are below. Two use the word “dilation”, and one does not. The remaining texts I have do not use the word “dilation”, but instead using “stretching”, “shrinking”, or “compressing”. I have also include some links to definitions on the internet at the end, several of which have slightly differing definitions than the one I have used in this post.

My local school district has been using “dilation” as their preferred term for a decade or so.

“Advanced Mathematical Concepts – Precalculus with Applications” published by McGraw Hill/Glencoe in 2007 (ISBN 0-07-868227-4) refers to dilations in a number of places starting on pg 88 and later, but really only defines the term in its Glossary on pg A73 “a transformation in which a figure is enlarged or reduced”.

“Elementary Linear Algebra” by Howard Anton, published by John Wiley & Sons in 1973 (ISBN 0-471-03247-6) describes a dilation on pg 186 as a transformation that “stretches each vector … by a factor of k”. This text uses the term “contraction” when a dilation factor between 0 and 1 is used, vs dilation when the factor is greater than 1.

“Functions, Statistics, and Trigonometry” by UCSMP, published by Scott Foresman Addison Wesley in 1998 (ISBN 0-673-45926-8) defines “scale change (in the plane)” on pg 921 as “The transformation that maps (x, y) to (ax, by), where a <>0 and b<>0 are constants”.

http://www.merriam-webster.com/dictionary/dilation

http://www.mathwords.com/d/dilation.htm

http://mathworld.wolfram.com/Dilation.html

7 comments

John says:

December 2, 2015 at 1:02 am

Dear Mr. Ford,

Thanks for this nice post. Can you please provide a reference (book/published paper) for the formal definition of ‘Function Dilation’?

1. Michael Paul Goldenberg says:
  
  December 2, 2015 at 2:01 pm
  
  Although in ordinary English, a dilation is a stretching, widening, or enlarging (when the doctor puts drops in your eyes to dilate them, your pupils get very large), in math, dilation is used for both stretching and shrinking. A better word might be “scaling,” since that carries the idea of both increasing and decreasing without having to specify which (we both “scale up” and “scale down”). I am not sure why you want a formal definition, but I suspect that if you search online math sources for “scaling” you’ll find what you want. In most functions, including polynomials, trig functions, exponential functions, etc., a coefficient that multiplies the output of a parent function after the input is mapped to some output depending on the function rule will have the effect of scaling (enlarging or shrinking, and possibly “flipping” around the x-axis if the sign of the coefficient is negative).
  
2. Whit Ford says:
  
  December 2, 2015 at 2:15 pm
  
  John,
  
  Some texts do not use the word dilation, choosing instead to use phrases such as “Stretching or Shrinking”, or “Stretching or Compressing”, or “Scale Change”.
  
  References to three texts on my bookshelf are below. Two use the word “dilation”, and one does not. The remaining texts I have do not use the word “dilation”, but instead using “stretching”, “shrinking”, or “compressing”. I have also include some links to definitions on the internet at the end, several of which have slightly differing definitions than the one I have used in this post.
  
  My local school district has been using “dilation” as their preferred term for a decade or so.
  
  “Advanced Mathematical Concepts – Precalculus with Applications” published by McGraw Hill/Glencoe in 2007 (ISBN 0-07-868227-4) refers to dilations in a number of places starting on pg 88 and later, but really only defines the term in its Glossary on pg A73 “a transformation in which a figure is enlarged or reduced”.
  
  “Elementary Linear Algebra” by Howard Anton, published by John Wiley & Sons in 1973 (ISBN 0-471-03247-6) describes a dilation on pg 186 as a transformation that “stretches each vector … by a factor of k”. This text uses the term “contraction” when a dilation factor between 0 and 1 is used, vs dilation when the factor is greater than 1.
  
  “Functions, Statistics, and Trigonometry” by UCSMP, published by Scott Foresman Addison Wesley in 1998 (ISBN 0-673-45926-8) defines “scale change (in the plane)” on pg 921 as “The transformation that maps (x, y) to (ax, by), where a <>0 and b<>0 are constants”.
  
  http://www.merriam-webster.com/dictionary/dilation
  
  http://www.mathwords.com/d/dilation.htm
  
  http://mathworld.wolfram.com/Dilation.html
  
bobbi gold says:

February 20, 2016 at 10:05 pm

I think from the explanation given for the original equation “g(x)=3(x-1)^2+5”, it should read “g(x)=3(x+1)^2+5” for there to be a +1 vertical translation.

1. Whit Ford says:
  
  February 20, 2016 at 11:07 pm
  
  The change you made in the equation affects its Horizontal translation, not its Vertical translation as you stated. See another post of mine for an explanation of why horizontal translations work the way they do: https://mathmaine.wordpress.com/2010/05/27/function-translations/
  
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Martin Mitchel says:

January 29, 2023 at 4:19 am

Wonderful explanation. I loved your elastic graph plane image. It really helped me picture what was going on.