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.
Definition: Dilation in the vertical direction
Consider a function,drawn on the plane. We stretch in the vertical direction by a scale factor of,cause the transformation.Also, the zeros of the function remain unchanged, as do the coordinates of any inflection points. The intersection value and coordinate of any inflection point are multiplied by the scale factor.
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.
Example 1: Expressing vertical expansions using function notation
The function is stretched vertically by a scale factor of.write about,the equation of the transformed function.
Stretching a function vertically by a scale factor of gives the transformation.Since the specified scale factor is,The new feature is.
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.
Definition: Dilation in the horizontal direction
Consider a function,drawn on the plane. Stretch horizontally by a scale factor or by creating the new function.The section value and coordinate of each inflection point remain unchanged. The zeros of the function are multiplied by the scale factor, as are the coordinates of any inflection point.
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.
Example 2: Expressing horizontal elongations using function notation
The function is extended horizontally by a scale factor of 2. Write it in the form of,the equation of the transformed function.
Stretching a function horizontally by a scale factor gives the transformation.Since the given scale factor is 2, the transform and therefore the new function is.
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.
Example 3: Identify the graph of a given function after a snippet
The figure shows the graph of.
Which of the following is the graph of?
The function represents a stretch in the vertical direction by a scale factor of,which means this is a compression. Since we are stretching the feature in the vertical direction, the coordinates of the keypoints will not be affected and we will focus our attention on the coordinates instead.
This means that we can ignore the roots of the function and instead focus on the part of,what seems to be the problem.If we were to graph the function,then we would divide the coordinate, giving the new section at that point.Of the charts provided, the only chart that takes this property into account is option (e), which means that this must be the correct choice. Note that the roots of this parcel are unaffected by the given span, indicating that we made the correct decision.
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.
Example 4: Expressing a dilation using function notation, showing the dilation graphically
The red graph in the figure represents the equation and the green graph represents the equation.As an espresso transformation.
We start by looking at the main points of the function.,drawn in red. First, the section is at the origin, so the period,which means it is also a root of.Stretching in the vertical or horizontal direction doesn't affect this point, so we'll ignore it below.
If we just look at the graph, we can see that the function has been stretched in the horizontal direction, which would indicate that the function has been stretched 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 when and,so at points and.However, in the new function,shown in green, we can see that there are roots if and,so at points and.This indicates that we have expanded by a scale factor of 2. The distance from the roots to the origin has doubled, which means that we have actually expanded the function by a factor of 2 in the horizontal direction.
We should review whether changes to the tipping point are consistent with this understanding. We can see that there is a local maximum of,which is to the left of the vertical axis and that there is a local minimum to the right of the vertical axis. now against,we can see that the coordinate of these inflection points seems to have bent while the coordinate has not changed. While this doesn't fully confirm what we found, as we can't be exact with the turning points on the graph, it certainly seems to agree with our solution. So we have the relation.
Example 5: Find the coordinates of a point on a curve after expanding the original function
The figure shows the graph of and the point.The point is a local maximum. Identify the corresponding local maximum for the transformation.
The transformation represents a stretch in the horizontal direction by a scale factor of.This halves the coordinate value of the keypoints without affecting the coordinates. In particular, the roots of em and,each has the coordinates and,which are also the two local minima of the function. If you look at the function,the coordinates change, giving the new roots at y,each with the coordinates and.With respect to the local maximum at the point,the coordinate is split in two and the coordinate remains intact, which means that the local maximum of is at this point.
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.
Example 6: Identify the graph of a given function after a snippet
The diagram shows the graph of the function for.
Which of the following options does the diagram show??
We observe that the function intersects the axis at points and that the function appears to cross the axis at points and.There are other points that are easy to identify and write in the form of coordinates. For example the points,y.
The stretch corresponds to a vertical compression by a factor of 3. This means that the feature must be “compressed” by a factor of 3 parallel to the axis. In terms of affecting the function's known coordinates, the coordinate of all annotated points is unaffected and their coordinate is divided by 3. Referring to the key points in the previous paragraph, these are converted as follows:,,,,y.
The only graph where the function passes through these coordinates is option (c). We can visually confirm that this function appears to be compressed by a factor of 3 in the vertical direction.
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.
- A feature can be scaled in the vertical direction by creating the new feature.
- When dilating a function in the vertical direction, the roots of the function remain unchanged, as do the coordinates of all inflection points.
- When scaling in the vertical direction, the intersection and coordinate value of any inflection point is also multiplied by the scale factor.
- When stretched in the vertical direction by a negative scale factor, the feature is reflected in the horizontal axis, in addition to the stretch/squeeze effect that occurs when the scale factor is not equal to unity. This transformation converts local minima to local maxima and vice versa.
- A function can be extended horizontally by a scale factor creating the new function.
- When stretched in the horizontal direction, the part of the function remains unchanged, as does the coordinate of any inflection point.
- When stretching in the horizontal direction, the roots of the function are stretched by the scale factor, as well as the coordinate of all inflection points.
- When stretched in the horizontal direction by a negative scale factor, the feature is reflected in the vertical axis, in addition to the stretch/squeeze effect that occurs when the scale factor is not equal to unity. This transformation does not affect the sorting of inflection points.
- We can expand in both directions, with a scale factor in the vertical direction and a scale factor in the horizontal direction, using the transform.
How do you describe dilation transformation? ›
A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. A dilation that creates a larger image is called an enlargement. A dilation that creates a smaller image is called a reduction. A dilation stretches or shrinks the original figure.What is the definition of dilation of a function? ›
A dilation is a stretching or shrinking about an axis caused by multiplication or division.What is the learning objective for dilation? ›
Use a scale factor to dilate an image. Experiment and verify visually the properties of dilations. Understand and verify that dilations do not alter angles. Understand and verify how dilations alter line and side length.
A dilation is a transformation which preserves the shape and orientation of the figure, but changes its size. The scale factor of a dilation is the factor by which each linear measure of the figure (for example, a side length) is multiplied.What is the difference between a transformation and a dilation? ›
Transformations change the location or orientation of an image but not the shape. Rigid motions are transformations that move an image, but do not change the size. The only transformation that is not a rigid motion is dilation. A dilation is a transformation that changes the size of a figure.What is dilation in simple words? ›
: the act or action of enlarging, expanding, or widening : the state of being dilated: such as. : the act or process of expanding (such as in extent or volume)What is dilation and example? ›
Dilation means changing the size of an object without changing its shape. The size of the object may be increased or decreased based on the scale factor. An example of dilation math is a square of side 5 units can be dilated to a square of side 15 units, but the shape of the square remains the same.How do you find the dilation? ›
Find the distance between a point on the preimage and the center of dilation. Multiply this length by the scale factor. The corresponding point on the image will be this distance away from the center of dilation in the same direction as the original point.How do you explain learning objectives? ›
Learning objectives are statements that describe significant and essential learning that learners have achieved, and can reliably demonstrate at the end of a course or program. In other words, learning objectives identify what the learner will know and be able to do by the end of a course or program.What are the 3 learning objectives examples? ›
These three types of learning include: Creating new knowledge (Cognitive) • Developing feelings and emotions (Affective) • Enhancing physical and manual skills (Psychomotor) Page 2 Learning objectives can also be scaffolded so that they continue to push student learning to new levels in any of these three categories.
How do you show dilation on a graph? ›
- Identify the center of dilation. ...
- Identify the original points of the polygon. ...
- Identify the scale factor . ...
- Multiply each original point of the polygon by the scale factor to get the new points. ...
- Plot the new points and connect the dots to get your dilated shape.
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.Does dilation mean bigger or smaller? ›
In geometry, dilating something just means changing its size. When shapes are dilated, only their size changes—not their location, orientation, or shape. In the image below, the shape gets bigger: its length and width are multiplied by 2.Does dilation mean expansion? ›
Dilation: The process of enlargement, stretching, or expansion. The word "dilatation" means the same thing. Both come from the Latin "dilatare" meaning "to enlarge or expand."How do you explain time dilation to a child? ›
An observer, B, on the ground sees the light moving at the same speed but it travels farther between bounces. The observer calculates the time between bounces to be longer than that measured by A. Thus B sees time passing more slowly on the fast-moving spaceship—an effect called time dilation.What happens during dilation? ›
Cervical effacement and dilation. During the first stage of labor, the cervix opens (dilates) and thins out (effaces) to allow the baby to move into the birth canal.What is a dilation for kids? ›
Dilation (sometimes incorrectly called dilatation) is a procedure that stretches abnormal tissues in and around the esophagus to help children with esophageal problems swallow normally.How to do a dilation step by step? ›
Two things are needed to dilate: an original shape and a scale factor k. Write down the coordinates of each point of the original shape and label them. To find the points of the new, dilated shape, simply multiply each of the original coordinates by k, then connect the dots!Is dilation adding or multiplying? ›
Dilations involve multiplication! Dilation with scale factor 2, multiply by 2.What are lesson objectives in a lesson plan? ›
Lesson Objective: The lesson objective states what students will know or be able to do at the end of the lesson. The strategies, materials, assignments, and assessments used in a lesson are determined by, and must align with, the lesson objective.
What are learning outcomes in lesson plan? ›
Learning outcomes are measurable statements that articulate at the beginning what students should know, be able to do, or value as a result of taking a course or completing a program (also called Backwards Course Design).How many are the lesson objectives? ›
There are three main types of learning objectives: cognitive, psychomotor, and affective.What are some examples of learning outcomes? ›
- Cognitive - knowledge related to a discipline. Example: Students will be able to identify major muscles groups.
- Skills and abilities - physical and intellectual skills related to a discipline. ...
- Affective - attitudes, behaviors and values related to a discipline.
Learning objective: Why the teacher is creating a learning activity. Example: This training session will discuss the new policy for reporting travel expenses. Learning outcome: What the learner will gain from the learning activity. Example: The learner understands how to properly report travel expenses.How to write a lesson plan? ›
- Identify the learning objectives. ...
- Plan the specific learning activities. ...
- Plan to assess student understanding. ...
- Plan to sequence the lesson in an engaging and meaningful manner. ...
- Create a realistic timeline. ...
- Plan for a lesson closure.
Definition: statements about what students should know/be able to do, what they might be asked to do to give evidence of learning, and how well they should be expected to know/do it. • Content standards refer to what students should know and be able to do.How do you describe the dilation of a graph? ›
Definition: Dilation in the Horizontal Direction
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 𝑥 -coordinates of any turning points.
A dilation is a type of transformation that enlarges or reduces a figure (called the preimage) to create a new figure (called the image). The scale factor, r, determines how much bigger or smaller the dilation image will be compared to the preimage.How can you describe the transformation? ›
Transformation of functions means that the curve representing the graph either "moves to left/right/up/down" or "it expands or compresses" or "it reflects". For example, the graph of the function f(x) = x2 + 3 is obtained by just moving the graph of g(x) = x2 by 3 units up.
How would you describe a transformation? ›
A transformation is a process that manipulates a polygon or other two-dimensional object on a plane or coordinate system. Mathematical transformations describe how two-dimensional figures move around a plane or coordinate system. A preimage or inverse image is the two-dimensional shape before any transformation.What is an example of dilation? ›
Dilation means changing the size of an object without changing its shape. The size of the object may be increased or decreased based on the scale factor. An example of dilation math is a square of side 5 units can be dilated to a square of side 15 units, but the shape of the square remains the same.How do you find dilation? ›
To find the scale factor for a dilation, we find the center point of dilation and measure the distance from this center point to a point on the preimage and also the distance from the center point to a point on the image. The ratio of these distances gives us the scale factor, as Math Bits Notebook accurately states.What is the rule for dilation in math? ›
To dilate a figure by a scale factor, multiply both the x-coordinate and the y-coordinate of each point by the scale factor. The new points are (4, -1), (6, -10), and (10, 8).