Explanation of the lesson: Transformations of functions: Dilation (2023)

In this explainer, we'll learn how to identify transformations of functions that involve horizontal and vertical stretching or squeezing.

When working with functions, we often want to stick to the graph to visualize and understand the overall behavior. Along with knowing specific information such as the roots, intercept, and any maximums or minimums, plotting the function can provide a complete picture of the exact known behavior, as well as a more general qualitative understanding. Once given or obtained an expression for a function, we are often interested in how this function can be written algebraically when subjected to geometric transformations such as rotations, reflections, translations and dilations.

In this discussion, we explore the concept of stretching, which is a general term for stretching or squeezing a feature (horizontally or vertically in this case) by a fixed scale factor. Geometrically, these transformations can sometimes be quite intuitive to visualize, although their algebraic interpretation can seem a bit counterintuitive, especially when extended in the horizontal direction. Consequently, we'll start by looking at vertical dilations before moving on to this slightly more complicated form of dilation.

Let's prove this definition by working with the square.We won't give an explanation here, but this function has two roots, one when and one when,with a section of,as well as a minimum in place.The graph of the function is shown below.

We now stretch the function in the vertical direction by a scale factor of 3. By our definition, this means that we have to apply the transform and use it to draw the function.

We could examine this new function and find that the location of the roots doesn't change. However, both the section and the minimum point have been shifted. The section value was multiplied by a scale factor of 3 and now has the value of.Although we do not specify the work here, the coordinate of the minimum also remains unchanged, although the new coordinate is three times the previous value, which means that this is the position of the new minimum point..This information is summarized in the graphic below, with the original resource shown in blue and the new resource in purple.

Likewise, we could have chosen to compress the feature by stretching it vertically by a scale factor between 0 and 1. For example, suppose we chose to stretch it vertically by a scale factor by applying the Morph.Then we would draw the function

This new function has the same roots as , but the value of -intercept is now.In addition, the position of the minimum point.This new feature, shown below in gold, overlaps with the previous rendering.

It's worth noting here that we only extend a function in the vertical direction by a positive scale factor. If we had chosen a negative scale factor, we would have done that too.reflectedthe function on the horizontal axis. This makes sense because it is well known that a function can be reflected about the horizontal axis by applying the transform.For example, stretching the function in the vertical direction by a scale factor can be considered first stretching the function with the transform,and then let it reflect further.This allows us to invert a function on the horizontal axis by stretching it in the vertical direction by a scale factor of.

Suppose we decide to stretch the given function in the vertical direction by a scale factor using the transform.Then we would have plotted the function

Again, the roots of this function do not change, but the --intercept has been multiplied by a scale factor of and now has a value of 4. The --coordinate of the minimum does not change, but the --coordinate has been multiplied by the scale factor. The new inflection point is,but it is now a local maximum as opposed to a local minimum. The new feature is shown in green below, overlapping the old one. We can see that the new function reflects the function on the horizontal axis.

Working with dilates in the horizontal direction may seem counterintuitive at first. While that's the case, we'll approach the treatment of dilations in the horizontal direction through the same framework as dilations in the vertical direction, and we'll discuss the implications at key points like the roots, intersections, and inflection points of the function we're working on. We'll start with a relevant definition and then demonstrate these changes by referencing the same quadratic function we used earlier.

We'll use the same function as before to understand dilations in the horizontal direction. As a reminder, we had the quadratic function,Her chart is below. We know that this function has two roots if and,also with a section,and a minimum point with the coordinate.

Let's first demonstrate the effects of dilation in the horizontal direction. We will choose an arbitrary scale factor of 2 using the transformation,and our definition implies that we must plot the function

If we were to analyze this function, we would find that the -intercept does not change and the -coordinate of the minimum is not affected either. The roots of the original function were at and,and we can see that the roots of the new function have been multiplied by the scale factor and are at and respectively. It's hard to see in the graph, but the coordinate of the minimum point has also been multiplied by the scale factor, which means that the minimum point now has the coordinate,considering it was for the original function.This information is summarized in the graph below, where the original function is shown in blue and the expanded function in purple.

Let's now look deeper into the previous definition by extending the function by a scale factor, which is between 0 and 1, and in this case we choose the scale factor.To create this effect of stretching the original function, we use the transform,which means we must draw the function

In this new function, the intersection and the coordinate of the inflection point are not affected. However, we could deduce that the value of the roots has halved, where the roots are now in y.In addition, the coordinate of the inflection point has also been halved, which means that the new location is.This is summarized in the graphic below, albeit not so clearly, where the new feature is represented in gold and overlays the old graphic.

In our last demo, we showed the effects of dilation in the horizontal direction by a negative scale factor. As with dilation in the vertical direction, we assume that there is a reflection, but this time on the vertical axis instead of the horizontal one. We will choose to extend the function in the horizontal direction by a scale factor of,that require transformation.Then we would draw the following function:

This new feature has the same section as,and the coordinate of the inflection point is not changed by this stretch. However, the roots of the new function have been multiplied by and are now at y,Whereas previously they were in respectively. The coordinate of the inflection point has also been multiplied by the scale factor and the new position of the inflection point is in.We draw the graph of the dilated function below where we can see the reflection effect on the vertical axis combined with the stretching effect. For clarity, we've simply drawn the original feature in blue and the new feature in purple.

As we mentioned earlier, it can be useful to understand dilations in terms of the impact they have on key points in a function, such as the B. section, the roots, and the positions of inflection points. If this information is known accurately, it is usually sufficient to infer the specific dilation without further examination. If you only work with a stretch in the vertical direction, the keypoint coordinates are not affected. If we only work with a section in the horizontal direction, the coordinates remain unchanged. We'll use this approach in the remaining examples of this explainer, always dilating only in the vertical or horizontal direction.

The next question gives a fairly typical example of graphical transformations, where a given slice is graphed and then we are asked to determine the exact algebraic transformation that represents it. While the question style is a bit more advanced than the previous example, the main focus remains basically the same. By paying attention to the behavior of key points, we will see that we can quickly deduce this information with little further investigation.

This explainer has worked so far with functions that were continuous when defined on the real axis, with all the "smooth" behaviors even when they were complicated. This need not be the case, instead we can work with a function that is not continuous or piecewise. In these situations, it's not entirely correct to use terminology like "intersection" or "root", because those terms are generally reserved for use with continuous functions. However, the principles still apply and we can continue with these themes by pointing out some key points and the effects they will have on vertical or horizontal sections.

In this explanatory work only with dilations exclusively in the vertical axis or in the horizontal axis; we do not consider a dilation that occurs simultaneously in both directions. Such transformations can be difficult to visualize even with the help of accurate graphing tools, especially when one of the scale factors is negative (meaning that one of the two involves axis symmetry). However, the result is actually quite easy to formulate. Suppose there is a function and we want to extend it by a scale factor in the vertical direction and a scale factor in the horizontal direction. Thus, we would obtain the new function due to the transformation

In many ways, our work so far on this explainer can be summarized with the following result, which describes the effect of simultaneous dilation on both axes. Suppose we take any coordinate on the graph of this new function that we are going to label.Now take the original function and extend it by a scale factor in the vertical direction and a scale factor in the horizontal direction to get a new function.So the point is on the graph of.This result generalizes the previous results to specific points such as intersections, roots and inflection points.