Question

the figure below is dilated with the center of dilation at the origin and a scale factor of 1/4 and then translated up 2 units. what are the coordinates of the image of point d after these transformations?

Explanation:

1. First, assume the coordinates of point D are \((x,y)\). From the graph, we can see that the coordinates of point D are \((- 8,-8)\).

• Step1: Perform dilation
• The rule for dilation with the center of dilation at the origin \((0,0)\) and a scale - factor \(k\) is \((x,y)\to(kx,ky)\). Here, \(k = \frac{1}{4}\), \(x=-8\), and \(y = - 8\).
• Calculate the new \(x\) - coordinate after dilation: \(x_1=k\times x=\frac{1}{4}\times(-8)=-2\).
• Calculate the new \(y\) - coordinate after dilation: \(y_1=k\times y=\frac{1}{4}\times(-8)=-2\).
• Step2: Perform translation
• The rule for translation up \(2\) units is \((x,y)\to(x,y + 2)\).
• For the point \((x_1,y_1)=(-2,-2)\) after dilation, the new \(x\) - coordinate remains the same \(x_2=x_1=-2\).
• Calculate the new \(y\) - coordinate: \(y_2=y_1 + 2=-2 + 2=0\).

Answer:

\((-2,0)\)