Question

graph △fgh with vertices f(-2,2), g(-2,-4) and h(-4,-4) and its image after the similarity transformation. translation: (x,y) → (x+3,y+1) dilation: (x,y) → (2x,2y)

Explanation:

Step1: Apply Translation to F

For point \( F(-2, 2) \), use translation \((x,y)\to(x + 3,y + 1)\).
New \( x \): \(-2+3 = 1\), New \( y \): \(2 + 1=3\). So \( F'=(1,3) \).

Step2: Apply Translation to G

For point \( G(-2,-4) \), apply translation.
New \( x \): \(-2+3 = 1\), New \( y \): \(-4 + 1=-3\). So \( G'=(1,-3) \).

Step3: Apply Translation to H

For point \( H(-4,-4) \), apply translation.
New \( x \): \(-4+3=-1\), New \( y \): \(-4 + 1=-3\). So \( H'=(-1,-3) \).

Step4: Apply Dilation to Translated Points

Dilation: \((x,y)\to(2x,2y)\)

• For \( F'(1,3) \): \( (2\times1,2\times3)=(2,6) \) (let's call this \( F'' \))
• For \( G'(1,-3) \): \( (2\times1,2\times(-3))=(2,-6) \) (let's call this \( G'' \))
• For \( H'(-1,-3) \): \( (2\times(-1),2\times(-3))=(-2,-6) \) (let's call this \( H'' \))

To graph:

1. Plot \( F(-2,2) \), \( G(-2,-4) \), \( H(-4,-4) \) and connect for \( \triangle FGH \).
2. Plot \( F'(1,3) \), \( G'(1,-3) \), \( H'(-1,-3) \) and connect for the translated triangle.
3. Plot \( F''(2,6) \), \( G''(2,-6) \), \( H''(-2,-6) \) and connect for the dilated (after translation) triangle.

(Note: The problem asks to graph, but the key is finding the transformed coordinates as above.)

Answer:

Translated vertices: \( F'(1, 3) \), \( G'(1, -3) \), \( H'(-1, -3) \); Dilated (after translation) vertices: \( F''(2, 6) \), \( G''(2, -6) \), \( H''(-2, -6) \) (Use these to graph the triangles).