Lesson Explainer: Function Transformations: Dilation Mathematics

In this explainer, we will learn how to identify function transformations involving horizontal and vertical stretches or compressions.

When working with functions, we are often interested in obtaining the graph as a means of visualizing and understanding the general behavior. Coupled with the knowledge of specific information such as the roots, the 𝑦-intercept, and any maxima or minima, plotting a graph of the function can provide a complete picture of the exact, known behavior as well as a more general, qualitative understanding. Once an expression for a function has been given or obtained, we will often be interested in how this function can be written algebraically when it is subjected to geometric transformations such as rotations, reflections, translations, and dilations.

In this explainer, we will investigate the concept of a dilation, which is an umbrella term for stretching or compressing a function (in this case, in either the horizontal or vertical direction) by a fixed scale factor. Geometrically, such transformations can sometimes be fairly intuitive to visualize, although their algebraic interpretation can seem a little counterintuitive, especially when stretching in the horizontal direction. Accordingly, we will begin by studying dilations in the vertical direction before building to this slightly trickier form of dilation.

Definition: Dilation in the Vertical Direction

Consider a function 𝑦=𝑓(π‘₯), plotted in the π‘₯𝑦-plane. We stretch it in the vertical direction by a scale factor of π‘Ž, causing the transformation 𝑓(π‘₯)β†’π‘Žπ‘“(π‘₯). Furthermore, the roots of the function are unchanged, as are the π‘₯-coordinates of any turning points. The value of the 𝑦-intercept, as well as the 𝑦-coordinate of any turning point, will be multiplied by the scale factor.

We will demonstrate this definition by working with the quadratic 𝑓(π‘₯)=π‘₯βˆ’π‘₯βˆ’2. We will not give the reasoning here, but this function has two roots, one when π‘₯=βˆ’1 and one when π‘₯=2, with a 𝑦-intercept of βˆ’2, as well as a minimum at the point ο€Ό12,βˆ’94. The plot of the function 𝑦=𝑓(π‘₯) is given below.

Now we will stretch the function in the vertical direction by a scale factor of 3. According to our definition, this means that we will need to apply the transformation 𝑓(π‘₯)β†’3𝑓(π‘₯) and hence sketch the function 𝑦=3𝑓(π‘₯)=3ο€Ήπ‘₯βˆ’π‘₯βˆ’2=3π‘₯βˆ’3π‘₯βˆ’6.

We could investigate this new function and we would find that the location of the roots is unchanged. However, both the 𝑦-intercept and the minimum point have moved. The value of the 𝑦-intercept has been multiplied by the scale factor of 3 and now has the value of βˆ’6. Although we will not give the working here, the π‘₯-coordinate of the minimum is also unchanged, although the new 𝑦-coordinate is thrice the previous value, meaning that the location of the new minimum point is ο€Ό12,βˆ’274. This information is summarized on the plot below, with the original function plotted in blue and the new function plotted in purple.

Equally, we could have chosen to compress the function 𝑓(π‘₯) by stretching it in the vertical direction by a scale factor of a number between 0 and 1. For example, suppose that we chose to stretch it in the vertical direction by a scale factor of 13 by applying the transformation 𝑓(π‘₯)β†’13𝑓(π‘₯). We would then plot the function 𝑦=13𝑓(π‘₯)=13ο€Ήπ‘₯βˆ’π‘₯βˆ’2=13π‘₯βˆ’13π‘₯βˆ’23.

This new function has the same roots as 𝑓(π‘₯) but the value of the 𝑦-intercept is now βˆ’23. Furthermore, the location of the minimum point is ο€Ό12,βˆ’34. This new function is plotted below in gold, superimposed over the previous plot.

At this point it is worth noting that we have only dilated a function in the vertical direction by a positive scale factor. Had we chosen a negative scale factor, we also would have reflected the function in the horizontal axis. This makes sense, as it is well-known that a function 𝑓(π‘₯) can be reflected in the horizontal axis by applying the transformation 𝑓(π‘₯)β†’βˆ’π‘“(π‘₯). For example, stretching the function in the vertical direction by a scale factor of βˆ’2 can be thought of as first stretching the function with the transformation 𝑓(π‘₯)β†’2𝑓(π‘₯), and then reflecting it by further letting 𝑓(π‘₯)β†’βˆ’π‘“(π‘₯). This allows us to think about reflecting a function in the horizontal axis as stretching it in the vertical direction by a scale factor of βˆ’1.

Suppose that we had decided to stretch the given function 𝑓(π‘₯) by a scale factor of βˆ’2 in the horizontal direction by using the transformation 𝑓(π‘₯)β†’βˆ’2𝑓(π‘₯). Then, we would have been plotting the function 𝑦=βˆ’2𝑓(π‘₯)=βˆ’2ο€Ήπ‘₯βˆ’π‘₯βˆ’2=βˆ’2π‘₯+2π‘₯+4.

Once again, the roots of this function are unchanged, but the 𝑦-intercept has been multiplied by a scale factor of βˆ’2 and now has the value 4. The π‘₯-coordinate of the minimum is unchanged, but the 𝑦-coordinate has been multiplied by the scale factor. The new turning point is ο€Ό12,92, but this is now a local maximum as opposed to a local minimum. The new function is plotted below in green and is overlaid over the previous plot. We can see that the new function is a reflection of the function 𝑦=2𝑓(π‘₯) in the horizontal axis.

Example 1: Expressing Vertical Dilations Using Function Notation

The function 𝑦=𝑓(π‘₯) is stretched in the vertical direction by a scale factor of 12. Write, in terms of 𝑓(π‘₯), the equation of the transformed function.

Answer

Stretching a function in the vertical direction by a scale factor of π‘Ž will give the transformation 𝑓(π‘₯)β†’π‘Žπ‘“(π‘₯). Since the given scale factor is 12, the new function is 𝑦=𝑓(π‘₯)2.

At first, working with dilations in the horizontal direction can feel counterintuitive. Much as this is the case, we will approach the treatment of dilations in the horizontal direction through much the same framework as the one for dilations in the vertical direction, discussing the effects on key points such as the roots, the 𝑦-intercepts, and the turning points of the function that we are interested in. We will begin with a relevant definition and then will demonstrate these changes by referencing the same quadratic function that we previously used.

Definition: Dilation in the Horizontal Direction

Consider a function 𝑦=𝑓(π‘₯), plotted in the π‘₯𝑦-plane. We stretch it in the horizontal direction by a scale factor of π‘Ž by creating the new function 𝑓(π‘₯)→𝑓1π‘Žπ‘₯. The value of the 𝑦-intercept, as well as the 𝑦-coordinate of any turning point, will be unchanged. The roots of the function are multiplied by the scale factor, as are the π‘₯-coordinates of any turning points.

We will use the same function as before to understand dilations in the horizontal direction. As a reminder, we had the quadratic function 𝑓(π‘₯)=π‘₯βˆ’π‘₯βˆ’2, the graph of which is below. We know that this function has two roots when π‘₯=βˆ’1 and π‘₯=2, also having a 𝑦-intercept of βˆ’2, and π‘₯𝑦-plane a minimum point with the coordinate ο€Ό12,βˆ’94.

We will first demonstrate the effects of dilation in the horizontal direction. We will choose an arbitrary scale factor of 2 by using the transformation 𝑓(π‘₯)→𝑓12π‘₯, and our definition implies that we should then plot the function 𝑦=𝑓12π‘₯=ο€»π‘₯2ο‡βˆ’ο€»π‘₯2ο‡βˆ’2=14π‘₯βˆ’12π‘₯βˆ’2.

If we were to analyze this function, then we would find that the 𝑦-intercept is unchanged and that the 𝑦-coordinate of the minimum point is also unaffected. The roots of the original function were at π‘₯=βˆ’1 and π‘₯=2, and we can see that the roots of the new function have been multiplied by the scale factor and are found at π‘₯=βˆ’2 and π‘₯=4 respectively. It is difficult to tell from the diagram, but the π‘₯-coordinate of the minimum point has also been multiplied by the scale factor, meaning that the minimum point now has the coordinate ο€Ό1,βˆ’94, whereas for the original function it was ο€Ό12,βˆ’94. This information is summarized in the diagram below, where the original function is plotted in blue and the dilated function is plotted in purple.

We will now further explore the definition above by stretching the function by a scale factor that is between 0 and 1, and in this case we will choose the scale factor 12. To create this dilation effect from the original function, we use the transformation 𝑓(π‘₯)→𝑓1π‘₯, meaning that we should plot the function 𝑦=𝑓1π‘₯=𝑓(2π‘₯)=(2π‘₯)βˆ’(2π‘₯)βˆ’2=4π‘₯βˆ’2π‘₯βˆ’2.

In this new function, the 𝑦-intercept and the 𝑦-coordinate of the turning point are not affected. However, we could deduce that the value of the roots has been halved, with the roots now being at π‘₯=βˆ’12 and π‘₯=1. Additionally, the π‘₯-coordinate of the turning point has also been halved, meaning that the new location is ο€Ό14,βˆ’94. This is summarized in the plot below, albeit not with the greatest clarity, where the new function is plotted in gold and overlaid over the previous plot.

In our final demonstration, we will exhibit the effects of dilation in the horizontal direction by a negative scale factor. As with dilation in the vertical direction, we anticipate that there will be a reflection involved, although this time in the horizontal axis instead of the vertical axis. We will choose to dilate the function in the horizontal direction by a scale factor of βˆ’2, which will require the transformation 𝑓(π‘₯)→𝑓1βˆ’2π‘₯. We would then plot the following function: 𝑦=𝑓1βˆ’2π‘₯=π‘“ο€Όβˆ’12π‘₯=ο€Όβˆ’12π‘₯οˆβˆ’ο€Όβˆ’12π‘₯οˆβˆ’2=14π‘₯+12π‘₯βˆ’2.

This new function has the same 𝑦-intercept as 𝑓(π‘₯), and the 𝑦-coordinate of the turning point is not altered by this dilation. However, the roots of the new function have been multiplied by βˆ’2 and are now at π‘₯=2 and π‘₯=βˆ’4, whereas previously they were at π‘₯=βˆ’1 and π‘₯=2 respectively. The π‘₯-coordinate of the turning point has also been multiplied by the scale factor and the new location of the turning point is at ο€Όβˆ’1,βˆ’94. We have plotted the graph of the dilated function below, where we can see the effect of the reflection in the vertical axis combined with the stretching effect. For the sake of clarity, we have only plotted the original function 𝑓(π‘₯) in blue and the new function in purple.

Example 2: Expressing Horizontal Dilations Using Function Notation

The function 𝑦=𝑓(π‘₯) is stretched in the horizontal direction by a scale factor of 2. Write, in terms of 𝑓(π‘₯), the equation of the transformed function.

Answer

Stretching a function in the horizontal direction by a scale factor of π‘Ž will give the transformation 𝑓(π‘₯)→𝑓1π‘Žπ‘₯. Since the given scale factor is 2, the transformation is 𝑓(π‘₯)→𝑓12π‘₯ and hence the new function is 𝑦=𝑓π‘₯2.

As we have previously mentioned, it can be helpful to understand dilations in terms of the effects that they have on key points of a function, such as the 𝑦-intercept, the roots, and the locations of any turning points. If this information is known precisely, then it will usually be enough to infer the specific dilation without further investigation. If we are working exclusively with a dilation in the vertical direction, then the π‘₯-coordinates of any key points will be unaffected. Similarly, if we are working exclusively with a dilation in the horizontal direction, then the 𝑦-coordinates will be unaffected. We will use this approach throughout the remainder of the examples in this explainer, where we will only ever be dilating in either the vertical or the horizontal direction.

Example 3: Identifying the Graph of a Given Function Following a Dilation

The figure shows the graph of 𝑦=𝑓(π‘₯).

Which of the following is the graph of 𝑦=12𝑓(π‘₯)?

Answer

The function 𝑦=12𝑓(π‘₯) represents a dilation in the vertical direction by a scale factor of 12, meaning that this is a compression. Given that we are dilating the function in the vertical direction, the π‘₯-coordinates of any key points will not be affected, and we will give our attention to the 𝑦-coordinates instead.

This means that we can ignore the roots of the function, and instead we will focus on the 𝑦-intercept of 𝑓(π‘₯), which appears to be at the point (0,2). If we were to plot the function 𝑦=12𝑓(π‘₯), then we would be halving the 𝑦-coordinate, hence giving the new 𝑦-intercept at the point (0,1). From the graphs given, the only graph that respects this property is option (e), meaning that this must be the correct choice. Note that the roots of this graph are unaffected by the given dilation, which gives an indication that we have made the correct choice.

The next question gives a fairly typical example of graph transformations, wherein a given dilation is shown graphically and then we are asked to determine the precise algebraic transformation that represents this. Much as the question style is slightly more advanced than the previous example, the main approach is largely unchanged. By paying attention to the behavior of the key points, we will see that we can quickly infer this information with little other investigation.

Example 4: Expressing a Dilation Using Function Notation Where the Dilation Is Shown Graphically

The red graph in the figure represents the equation 𝑦=𝑓(π‘₯) and the green graph represents the equation 𝑦=𝑔(π‘₯). Express 𝑔(π‘₯) as a transformation of 𝑓(π‘₯).

Answer

We will begin by noting the key points of the function 𝑓(π‘₯), plotted in red. Firstly, the 𝑦-intercept is at the origin, hence the point (0,0), meaning that it is also a root of 𝑓(π‘₯). Dilating in either the vertical or the horizontal direction will have no effect on this point, so we will ignore it henceforth.

Just by looking at the graph, we can see that the function has been stretched in the horizontal direction, which would indicate that the function has been dilated in the horizontal direction. To make this argument more precise, we note that in addition to the root at the origin, there are also roots of 𝑓(π‘₯) when π‘₯=βˆ’2 and π‘₯=1, hence being at the points (βˆ’2,0) and (1,0). However, in the new function 𝑔(π‘₯), plotted in green, we can see that there are roots when π‘₯=βˆ’4 and π‘₯=2, hence being at the points (βˆ’4,0) and (2,0). This indicates that we have dilated by a scale factor of 2. The distance from the roots to the origin has doubled, which means that we have indeed dilated the function in the horizontal direction by a factor of 2.

We should double check that the changes in any turning points are consistent with this understanding. We can see that there is a local maximum of 𝑓(π‘₯), which is to the left of the vertical axis, and that there is a local minimum to the right of the vertical axis. Now comparing to 𝑔(π‘₯), we can see that the π‘₯-coordinate of these turning points appears to have doubled, whereas the 𝑦-coordinate has not changed. Although this does not entirely confirm what we have found, since we cannot be accurate with the turning points on the graph, it certainly looks as though it agrees with our solution. Therefore, we have the relationship 𝑔(π‘₯)=𝑓π‘₯2.

Example 5: Finding the Coordinates of a Point on a Curve After the Original Function Is Dilated

The figure shows the graph of 𝑦=𝑓(π‘₯) and the point 𝐴. The point 𝐴 is a local maximum. Identify the corresponding local maximum for the transformation 𝑦=𝑓(2π‘₯).

Answer

The transformation 𝑦=𝑓(2π‘₯) represents a dilation in the horizontal direction by a scale factor of 12. This will halve the value of the π‘₯-coordinates of the key points, without affecting the 𝑦-coordinates. In particular, the roots of 𝑓(π‘₯) at π‘₯=1 and π‘₯=3, respectively, have the coordinates (1,0) and (3,0), which also happen to be the two local minimums of the function. When considering the function 𝑦=𝑓(2π‘₯), the π‘₯-coordinates will change and hence give the new roots at π‘₯=12 and π‘₯=32, which will, respectively, have the coordinates ο€Ό12,0 and ο€Ό32,0. Regarding the local maximum at the point (2,1), the π‘₯-coordinate will be halved and the 𝑦-coordinate will be unaffected, meaning that the local maximum of 𝑦=𝑓(2π‘₯) will be at the point (1,1).

This explainer has so far worked with functions that were continuous when defined over the real axis, with all behaviors being β€œsmooth,” even if they are complicated. This does not have to be the case, and we can instead work with a function that is not continuous or is otherwise described in a piecewise manner. In these situations, it is not quite proper to use terminology such as β€œintercept” or β€œroot,” since these terms are normally reserved for use with continuous functions. However, the principles still apply and we can proceed with these problems by referencing certain key points and the effects that these will experience under vertical or horizontal dilations.

Example 6: Identifying the Graph of a Given Function following a Dilation

The diagram shows the graph of the function 𝑦=𝑓(π‘₯) for βˆ’3≀π‘₯≀3.

Which of the following shows the graph of 𝑦=13𝑓(π‘₯)?

Answer

We note that the function intersects the 𝑦-axis at the point (0,βˆ’3) and that the function appears to cross the π‘₯-axis at the points ο€Όβˆ’32,0 and ο€Ό52,0. There are other points which are easy to identify and write in coordinate form. For example, the points (βˆ’3,3),(1,βˆ’3), and (3,1).

The dilation 𝑦=13𝑓(π‘₯) corresponds to a compression in the vertical direction by a factor of 3. This means that the function should be β€œsquashed” by a factor of 3 parallel to the 𝑦-axis. In terms of the effects on known coordinates of the function, any noted points will have their π‘₯-coordinate unaffected and their 𝑦-coordinate will be divided by 3. Referring to the key points in the previous paragraph, these will transform to the following, respectively: ο€Όβˆ’32,0, ο€Ό52,0, (βˆ’3,1), (1,βˆ’1), and ο€Ό3,13.

The only graph where the function passes through these coordinates is option (c). We can confirm visually that this function does seem to have been squished in the vertical direction by a factor of 3.

In this explainer, we only worked with dilations that were strictly either in the vertical axis or in the horizontal axis; we did not consider a dilation that occurs in both directions simultaneously. Such transformations can be hard to picture, even with the assistance of accurate graphing tools, especially if either of the scale factors is negative (meaning that either involves a reflection about the axis). The result, however, is actually very simple to state. Suppose that there is a function 𝑦=𝑓(π‘₯) and we wish to dilate this by a scale factor of π‘Ž in the vertical direction and a scale factor of 𝑏 in the horizontal direction. Then, we would obtain the new function by virtue of the transformation 𝑓(π‘₯)βŸΆπ‘Žπ‘“ο€Ό1𝑏π‘₯.

In many ways, our work so far in this explainer can be summarized with the following result, which describes the effect of a simultaneous dilation in both axes. Suppose that we take any coordinate on the graph of this the new function, which we will label (π‘₯,𝑦). Now take the original function and dilate it by a scale factor of π‘Ž in the vertical direction and a scale factor of 𝑏 in the horizontal direction to give a new function 𝑔(π‘₯)=π‘Žπ‘“ο€Ό1𝑏π‘₯. Then, the point (𝑏π‘₯,π‘Žπ‘¦) lays on the graph of 𝑔(π‘₯). This result generalizes the earlier results about special points such as intercepts, roots, and turning points.

Key Points

  • A function 𝑓(π‘₯) can be dilated in the vertical direction by a scale factor of π‘Ž by creating the new function 𝑓(π‘₯)β†’π‘Žπ‘“(π‘₯).
  • When dilating a function in the vertical direction, the roots of the function are unchanged, as are the π‘₯-coordinates of any turning points.
  • When dilating in the vertical direction, the value of the 𝑦-intercept, as well as the 𝑦-coordinate of any turning point, will also be multiplied by the scale factor.
  • When dilating in the vertical direction by a negative scale factor, the function will be reflected in the horizontal axis, in addition to the stretching/compressing effect that occurs when the scale factor is not equal to negative one. This transformation will turn local minima into local maxima, and vice versa.
  • A function 𝑓(π‘₯) can be dilated in the horizontal direction by a scale factor of π‘Ž by creating the new function 𝑓(π‘₯)→𝑓1π‘Žπ‘₯.
  • When dilating in the horizontal direction, the 𝑦-intercept of the function is unchanged, as is the 𝑦-coordinate of any turning point.
  • When dilating in the horizontal direction, the roots of the function are stretched by the scale factor, as will be the π‘₯-coordinate of any turning points.
  • When dilating in the horizontal direction by a negative scale factor, the function will be reflected in the vertical axis, in addition to the stretching/compressing effect that occurs when the scale factor is not equal to negative one. This transformation does not affect the classification of turning points.
  • We can dilate in both directions, with a scale factor of π‘Ž in the vertical direction and a scale factor of 𝑏 in the horizontal direction, by using the transformation 𝑓(π‘₯)β†’π‘Žπ‘“ο€Ό1𝑏π‘₯.

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