To evaluate the student's answer, we need to follow a detailed, step-by-step approach to determine the correct y-coordinate after the dilation.
1. Identify the original point and the mapped point:
- The original point is [tex]\((4, -6)\)[/tex].
- The mapped point after dilation is [tex]\((12, y)\)[/tex], where [tex]\(y\)[/tex] is unknown.
2. Determine the scale factor of the dilation:
- The dilation has a center at [tex]\((0,0)\)[/tex].
- To find the scale factor [tex]\(k\)[/tex], use the x-coordinates of the points:
[tex]\( k = \frac{\text{mapped x-coordinate}}{\text{original x-coordinate}} = \frac{12}{4} = 3 \)[/tex].
3. Apply the scale factor to find the new y-coordinate:
- Use the scale factor [tex]\(k = 3\)[/tex] to compute the y-coordinate:
[tex]\[
y_{\text{computed}} = \text{original y-coordinate} \times k = -6 \times 3 = -18
\][/tex]
4. Compare the computed y-coordinate with the student's y-coordinate:
- The student thought [tex]\(y\)[/tex] is [tex]\(-2\)[/tex].
- Clearly, [tex]\(-2\)[/tex] is not equal to [tex]\(-18\)[/tex].
5. Identify the type of mistake the student made:
- Since the student obtained [tex]\(-2\)[/tex] instead of [tex]\(-18\)[/tex], let's analyze the possible errors:
- If the student incorrectly calculated the scale factor to be [tex]\(-2\)[/tex], then the x-coordinate would have been [tex]\((4 \times -2) = -8\)[/tex], which isn't the case since the correct x-coordinate is [tex]\(12\)[/tex]. So, option B is incorrect.
- If the student incorrectly divided by the scale factor instead of multiplying, let's check:
[tex]\[
\text{Incorrect operation} = \text{original y-coordinate} \div k = -6 \div 3 = -2
\][/tex]
This matches the student's answer. Thus, the student divided by the scale factor instead of multiplying it.
- The other options (adding the scale factor or the student being correct) are clearly not consistent with the computations.
Hence, the correct evaluation is:
C. The student incorrectly divided by the scale factor instead of multiplying by it.
