Function Transformations: Dilation

This post assumes you already familiar with analyzing function translations. Even if you are, reading Function Transformations: Translation may be a useful introduction, as it uses this same approach to understanding transformations. Note that
– Translations move a graph, but do not change its shape
– Dilations change the shape of a graph, often causing “movement” in the process

The red curve in the image above is a “transformation” of the green one. It has been “dilated” (or stretched) horizontally by a factor of 3. A dilation is a stretching or shrinking about an axis caused by multiplication or division. You can think of a dilation as the result of drawing a graph on rubberized paper, stapling an axis in place, then either stretching the graph away from the axis in both directions, or squeezing it towards the axis from both sides.

Transformations are often easiest to analyze by focusing on how the location of specific  points on the curve have changed. In the image above, the point A_1 on the green curve “corresponds” to point A_2 on the red curve. By this we mean that the transformation has moved point A_1 on the green graph to be at A_2 on the red graph.

Horizontal Dilations

In looking at the coordinates of the two corresponding points identified in the graph above, you can see that the y-coordinate has not changed and the x-coordinate has been stretched to be three times further away from the y-axis… changing the shape of the curve in the process. If you were to examine any other pair of corresponding points, you would see the exact same scaling factor at work. This can be described algebraically by the equation:

\begin{cases}3\cdot x_1~=~x_2~~~~~(1)\\*y_1~=~y_2\end{cases}

where x_1 is the starting x-coordinate from a point on the green curve, and x_2 is the dilated x-coordinate from the corresponding point on the red curve.

The equations above can be read as “three times an x-coordinate on the green graph produces the transformed x-coordinate on the red graph, while the y-coordinates will be the same.”

Another way of describing this situation is to say “the green curve has been dilated horizontally by a factor of 3, and vertically by a factor of 1.”

Note that every dilation equation will involve the multiplication or division of the original coordinate by a constant: the dilation factor. Addition and subtraction cause translations to occur, while multiplication or division cause dilations to occur.

Have you noticed that one value of x was not changed by this horizontal dilation? Which x_1 value is identical to its transformed x_2 value based on the equation above? Looking at the graph may help: what point do both the green and red curves have in common?

Points that lie on the axis perpendicular to the direction of dilation will not move. Since their coordinate on the other axis is zero, multiplying this value by the dilation factor will leave the coordinate at zero, and the dilated point will be the same as the original point.

The green curve above is the graph of the equation:

y_1=(x_1)^2~~~~~~~(2)

To find the equation of the translated red curve, sole the translation equations (1) for x_1 and y_1, then substitute for those variables in equation (2). y_1 is already solved for, so we only need to solve for x_1:

3\cdot x_1~=~x_2~~~~~(1)\\*~\\*x_1~=~\dfrac{1}{3}x_2

and the system of transformation equations is now ready for substitution:

\begin{cases}x_1~=~\dfrac{1}{3}x_2\\*y_1~=~y_2\end{cases}

Substituting this system into (2) produces:

y_1=(x_1)^2~~~~~~(2)~~Green\\*~\\*y_2=(\dfrac{1}{3}x_2)^2~~~~~~~~Red

Notice that the process of solving for x_1 before substituting causes the horizontal dilation factor to move to the other side of the equation, becoming the reciprocal of the original factor. A horizontal dilation by a factor of 3 causes the original (x_1) to become (\dfrac{1}{3}x_2) in the transformed equation.

When dilation factors are coefficients of the variable they affect (as opposed to on the other side of the equation), they will be the reciprocal of the dilation factor.

Vertical Dilations

Dilations-Y

Look at the coordinates of the corresponding A points on the green and red graphs above. The x-coordinate has not changed, but the y-coordinate has been stretched to be three times further away from the x-axis. If you were to examine any other pair of corresponding points, you would see the exact same relationship. This can be described algebraically by the system of transformation equations:

\begin{cases}x_1~=~x_2\\*3\cdot y_1~=~y_2~~~~~(3)\end{cases}

where y_1 is the starting y-coordinate from a point on the green curve, and y_2 is the dilated y-coordinate from the corresponding point on the red curve. The x coordinates do not change, and thus are set equal to one another.

Note once again that one point does not move in this dilation… the one shared by both graphs: the origin. Since it is on the x-axis, a vertical dilation does not move it.

The green curve is the graph of the equation:

y_1~=~(x_1)^2~~~~~~~(2)

To find the equation of the dilated curve, substitute the equations (3) above into the equation of the curve (2). To do so, we’ll need to solve equation (3) for y_1:

3\cdot y_1~=~y_2~~~~~(3)\\*~\\*y_1~=~\dfrac{1}{3}y_2

and the system of transformation equations becomes:

\begin{cases}x_1~=~x_2\\*y_1~=~\dfrac{1}{3}y_2\end{cases}

Substitute the results into (2) to produce an equation for y_2:

(y_1)~=~(x_1)^2~~~~~~(2)~~Green\\*~\\*(\dfrac{1}{3}y_2)~=~(x_2)^2~~~~~~~~Red

Notice that the process of solving for y_1 before substituting moves the vertical dilation amount to the other side of the equation, so that a vertical dilation by a factor of three becomes (\dfrac{1}{3}y_2) in the transformed equation. If this equation is then solved for y_2, we get:

(\dfrac{1}{3}y_2)~=~(x_2)^2~~~~~~~~Red\\*~\\*y_2~=~3(x_2)^2

The last equation above can be interpreted as the original result of the function, x^2, with every result stretched to be three times farther away from the x-axis. To summarize, dilation factors appear:

  • as the reciprocal of the stretch/compress factor when they are a coefficient of the variable they affect
  • as the exact stretch/compress factor when on the other side of the equation from, and not a coefficient of, the variable they affect

Negative Dilation Factors

A negative dilation factor will cause the graph to be reflected about an axis, as well as stretched or compressed.

For the vertical dilation example above, if the dilation factor had been -3 the resulting graph would have opened down (been reflected about the horizontal axis) and been stretched to have every point be three times further from the x-axis.

The example below includes a negative dilation factor.

Multiple Dilations

Dilation-XY

The image above shows the result of two dilations. The green curve has been dilated both horizontally and vertically to produce the red curve. Looking at the coordinates of the corresponding point A, you can see that it has been stretched horizontally by a factor of three, and vertically by a factor of negative two. Algebraically, this can be described by the system:

\begin{cases}~~3\cdot x_1~=~x_2\\*~~-2\cdot y_1~=~y_2\end{cases}

Looking at the two corresponding points from the graph above, (1,1) is the “A” point on the green curve, referred to generically as (x_1,y_1) in the system of translation equations above, and (3, -2) is the corresponding dilated point represented by (x_2, y_2) in the system of transformation equations above.

Solving both equations for the “original” (green) curve variables produces

\begin{cases}~~x_1~=~\dfrac{1}{3}x_2\\*~\\*~~y_1~=~\dfrac{-1}{2}y_2\end{cases}

and substituting both into the equation of the green curve leads to

~~~~~~~~y_1~=~(x_1)^2~~~~~~~(2)~~Green\\*~\\*(\dfrac{-1}{2}y_2)~=~(\dfrac{1}{3}x_2)^2~~~~~~~~~Red\\*~\\*~~~~~~~~~~~~or\\*~\\*~~~~~~~~y_2~=~~-2\cdot (\dfrac{1}{3}x_2)^2~~~Red

Note that dilation amounts appear:

  • as the reciprocal of the stretch/compress factor when they are a coefficient of the variable they affect
  • as the exact stretch/compress factor when on the other side of the equation from, and not a coefficient of, the variable they affect

Another Example

Dilation-XY2

The graph above shows a different green function than all the previous examples, in that the green curve does not intersect the x-axis. It does intercept the y-axis. The red curve is a dilation of the green, by a factor of 3 horizontally, and a factor of -2 vertically. You can use the corresponding points labelled A to verify these vertical and horizontal dilation factors.

In this example, as in previous ones, there is one point on the curve that does not move as a result of the horizontal dilation. The y-intercept of the green curve is not affected by the horizontal dilation because it lies on the y-axis, which perpendicular to the direction of horizontal dilation.

However, this is the first example in this article in which every point on the green curve is affected by the vertical dilation. This happens because the green curve never crosses the x-axis, and the vertical dilation factor stretches every green point to be twice as far away from the x-axis as it was on the green curve, so every point on the green curve moves as it is stretched away from the x-axis.

Notice that the red curve appears to have “moved” when compared to the green one, as the two curves do not share any points. Yet no translation was involved, because nothing was added or subtracted to the original coordinates. The dilation process, by stretching a curve away from an axis, can appear to “move” the curve as part of the stretching process… particularly when the curve does not cross the axis it is being stretched away from.

Dilations in More Complex Equations

When using substitution, all instances of the variable being substituted for must be replaced. The same applies when dilating a function. For example, if are working with a function that has multiple x terms, such as

y_1~=~(x_1)^2+3(x_1)+5

and dilating it horizontally by a factor of 4

4\cdot x_1~=~x_2\\*~\\*x_1~=~\dfrac{1}{4}x_2

then every instance of x_1 must have the dilation equation substituted in its place

y_2~=~(\dfrac{1}{4}x_2)^2+3(\dfrac{1}{4}x_2)+5

So, when analyzing one function to see if it a dilation of another, check that every instance of each variable has had the same transformation equation applied before asserting that the function is a transformation!

In this example,

y_2~=~(\dfrac{1}{4}x_2)^2+3(\dfrac{1}{7}x_2)+5

y2 does not describe a horizontal dilation of y1, because the “x” term did not have the same transformation equation substituted for it as the “x squared” term did.

Equivalent Dilations

In mathematics, it is often (but not always) possible to produce the same end result in different ways. When working with quadratic equations and using the approach described above, you may have wondered how to handle a situation such as:

y~=~(2x)^2

The above describes a horizontal dilation of y~=~x^2 by a factor of \dfrac{1}{2}, but if we expand and solve for x the equation becomes:

y~=~4(x^2)\\*~\\*\dfrac{1}{4}y~=~x

which describes a vertical dilation by a factor of 4. Are they both valid interpretations?

Since both of the above are algebraic manipulations of the same equation, they must both have the same graph.

Dilation-X or Y

Examine the graph of y~=~x^2 in green above, focusing on the point (1,1). Think about what happens to the x-coordinate of the point as the curve is dilated horizontally by a factor of \dfrac{1}{2} … it must move halfway to the y-axis, and become 0.5. Indeed, the point (0.5, 1) lies on the red curve.

Now think about that same (1, 1) point on the green curve, and what would happen to it if the curve is dilated vertically by a factor of 4. Its y-coordinate would become four times larger, moving the point to (1, 4)… which also lies on the red curve.

So both approaches end up being equivalent: they transform the entire green curve into the red one, one by squeezing horizontally, the other by stretching vertically.

Note that these equivalent dilations do not rely on the same dilation factor. The horizontal dilation factor gets squared, then is turned into its reciprocal to become the vertical dialation factor.

Note also that equivalent dilations will not exist for every curve. Asymmetrical curves generally won’t have two ways of producing a given dilation, so this is a bit of a special case.

Just as there are often multiple ways of describing a specific situation using words, math can often also describe something in multiple ways.

To apply your knowledge of both translations and dilations, check out the next article in this series: Using Corresponding Points to Determine Dilation Factors and Translation Amounts

Published by

Whit Ford

Math Tutor in Yarmouth, Maine

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