In mathematics, dilations are transformations that change the size of a figure while maintaining its shape. They are commonly used in geometry, especially when dealing with graphs. Generally, dilations are used to transform a figure by either expanding or shrinking it. I believe that understanding dilations is crucial for students who are taking algebra or geometry courses.
So, how do you calculate dilations? You need an original shape and a scale factor of k to calculate dilations. First, start by writing down the coordinates of every point of the original object and label them. To locate the points of the new, dilated shape, multiply each of the original coordinates by k, then connect each point.
In this post, I will provide step-by-step guidelines on how to do dilations on a graph. If you wonder how to do a reflection in geometry, I wrote a whole article with examples and steps by steps.
Understanding the Basics of Dilations
A dilation is a transformation that changes the size of a figure by a scale factor, which is a constant that is applied to both the x and y coordinates.
The scale factor can be greater than one (enlargement) or less than one (reduction). The center of dilation is the fixed point around which the figure is dilated.
Step 1: Identifying the Center of Dilation
The first step to dilating a graph is to identify the center of dilation. This can be done by looking for a point on the graph that serves as the fixed point for the dilation.
Once you have identified the center of dilation, draw a straight line through the center and the point you want to dilate. This line is known as the dilation line.
The center of dilation is the point that the figure will expand or shrink away from. It can be any point on the graph, but it is usually best to choose a point that makes sense for the figure you’re working with. For example, if you’re working with a square, then the center of dilation should be the center of the square.
Step 2: Finding the Scale Factor
The next step is to find the scale factor, which is the amount by which the figure will expand or shrink. To determine the scale factor, simply measure the distance from the center of dilation to any point on the figure. Then, measure the distance from the center of the dilation to the corresponding point on the dilated figure. Divide the two measurements to get the scale factor.
Scale Factor = Dilation image length divided by preimage length.
This ratio will give you the scale factor. If the ratio is greater than one, the figure is enlarged, and if it is less than one, the figure is reduced. In other words, the scale factor can be any number greater than 1, which will result in an expansion, or any number between 0 and 1, which will result in a shrinkage.
Step 3: Multiply the Coordinates by The Scale Factor
After determining the scale factor, the next step is to multiply each of the coordinates of the original figure by the scale factor.
If the scale factor is greater than 1, this will result in an expansion. If the scale factor is less than 1, this will result in a shrinkage.
Step 4: Graph the Dilated Figure
After you have multiplied the coordinates by the scale factor, you can now graph the dilated figure. Plot each of the new points on the graph and connect them to form the new figure.
Remember to label the new figure accordingly so that it’s easy to differentiate from the original figure.
Step 5: Checking Your Answers
The last step is to check your answers to make sure that you have correctly dilated the graph. You can check your answers by comparing the original and dilated graphs side by side and looking for any differences in size and shape.
If the graphs are identical, then you have successfully dilated the graph. If you want to learn more about dilations, I encourage you to check out Khan Academy and watch this video or the video below.
Wrapping Up
I believe that dilations are an important concept in mathematics, and understanding them is crucial for students who are taking algebra or geometry courses.
You can perform dilations on a graph by choosing the center of dilation, determining the scale factor, multiplying the coordinates by the scale factor, graphing the dilated figure, and practicing and reviewing regularly.
Knowing how to do dilations on a graph can help you solve a wide range of problems involving dilations, from simple geometry tasks to more complex data analysis.