Question

add ruler clear all show your work here center (2, - 1) and scale factor 3 center (3, 0) and scale factor 3 center (2, - 1) and scale factor $\frac{1}{3}$ center (3, 0) and scale factor $\frac{1}{3}$ center (2, - 1) and scale factor $\frac{5}{2}$ center (3, 0) and scale factor $\frac{5}{2}$

Explanation:

Step1: Recall dilation formula

For a dilation about a center \((h,k)\) with scale - factor \(a\), the transformation of a point \((x,y)\) to \((x',y')\) is given by \(x'=h + a(x - h)\) and \(y'=k + a(y - k)\). We can also use the property that if we know a point on the pre - image and its corresponding point on the image, we can find the center and scale factor.

Step2: Test the center \((2,-1)\)

Let's assume a point on the pre - image and its corresponding point on the image. If the center of dilation is \((h = 2,k=-1)\), and we take a point \((x,y)\) on the pre - image and \((x',y')\) on the image. Let's take a simple point on the pre - image and see if the scale factor calculation works. If we consider a point on the pre - image near the origin and its corresponding point on the image. The distance from the center \((2,-1)\) to a point on the pre - image and to the corresponding point on the image should give us the scale factor. By visual inspection, if we consider the relative distances of points from the center \((2,-1)\) on the pre - image and image, we find that the scale factor \(a = 3\). For example, if we consider a vector from the center \((2,-1)\) to a point on the pre - image and the vector from the center \((2,-1)\) to the corresponding point on the image, the length of the second vector is 3 times the length of the first vector.

Step3: Check other options

If we test the center \((3,0)\), the distances from the center \((3,0)\) to points on the pre - image and corresponding points on the image do not give a consistent scale factor of 3 or \(\frac{1}{3}\) or \(\frac{5}{2}\).

Answer:

Center \((2,-1)\) and scale factor \(3\)