This applet is designed to help you visualize how dilations affect object in the plane, and indeed, the plane itself. One way to visualize a dilation is to imagine that it is stretching the original plane out in all directions by the scale factor. So, a scale factor of 2 would stretch the plane in all directions from the center of dilation so that all distances are doubled. If you move the slider for k, the scale factor, you will see the square array of points dilate in response to the scale factor being changed. Think of this array as the plane the is being "stretched" by the dilation. You can click the "show triangle" box to plot a triangle (feel free to drag its points around to create different triangles) and see how the dilation affects the triangle.
Once you get the general feel, you can even drag the center of the dilation (point D in this applet) around to create a dilation whose center is NOT the origin. This is an important concept in the common core, and the main reason I created this applet. Try resetting the scale factor to 1 and the moving D to somewhere on the plane. Notice how when you change the scale factor all the points in the plan now stretch away from D (the NEW center of dilation) and not the origin as was previously the case. Play around with different scale factors, centers, and triangles to get a better feel of how all this works.
Finally, the "Stop at 1" button changes the range of your scale factor slider to go all the way down to zero. By clicking that button, you can scale by numbers between 0 and 1, which will pull all the points on the plane in towards the center of dilation. Enjoy!