Question

identify the center and scale factor of the dilation. the black shape is the pre - image and the blue shape is the image.

Explanation:

Step1: Determine the center of dilation

The center of dilation is the point that remains fixed during the dilation. By observing the graph, we can see that the point \((4,-4)\) is common to both the pre - image (black shape) and the image (blue shape). So, the center of dilation is \((4,-4)\).

Step2: Calculate the scale factor

Let's take a pair of corresponding sides. For example, consider a horizontal side of the pre - image and the image.
Take a point on the pre - image, say \((- 3,-2)\) and its corresponding point on the image. Let's assume we use the ratio of distances from the center of dilation.
Another way: If we consider the transformation of coordinates. Let \((x,y)\) be a point on the pre - image and \((x',y')\) be the corresponding point on the image. The formula for dilation about the center \((h,k)\) is \(x'=h + r(x - h)\) and \(y'=k + r(y - k)\), where \(r\) is the scale factor.
Let's take a non - center point. For example, take a vertex of the black triangle \((-3,-2)\) and a corresponding vertex of the blue triangle. Let's assume we use the vertical or horizontal distances from the center \((4,-4)\).
The distance from \((-3,-2)\) to \((4,-4)\) in the \(x\) - direction: \(d_{x1}=\vert-3 - 4\vert=7\), and for a corresponding point (say, if we consider the similar - shaped triangle) the distance from a point on the blue triangle (after dilation) in the \(x\) - direction from \((4,-4)\) is \(d_{x2}=14\).
The scale factor \(r=\frac{\text{distance from center of image point}}{\text{distance from center of pre - image point}}\).
If we use the formula of dilation. Let’s assume a general point \((x,y)\) and its image \((x',y')\) after dilation about \((h,k)=(4,-4)\).
Take a point \((-3,-2)\) (pre - image). Let the image of \((-3,-2)\) be \((x',y')\).
We know that \(x'=4+r(-3 - 4)\) and \(y'=-4+r(-2+4)\)
Another approach: Count the number of units from the center.
The pre - image (black triangle) and image (blue triangle). If we consider the length of a side. Suppose a side of the black triangle: if we count the number of units from the center \((4,-4)\) to a vertex of the black triangle (say moving horizontally or vertically).
Let’s assume we use the ratio of the lengths of corresponding sides.
The length of a side of the black triangle (pre - image) and the blue triangle (image). If we consider the fact that if we take two similar triangles (since dilation is a similarity transformation).
The scale factor \(r = 2\). We can check by looking at the distance from the center. For example, if a point on the pre - image is \(a\) units away from the center and the corresponding point on the image is \(2a\) units away from the center.

Answer:

Center: \((4,-4)\), Scale factor: \(2\)