Question

multiple-choice question
point q is the pre - image point. what are the coordinates of the image, q?
options: (-2, -2), (2, 2), (2, -2), (-2, 2)
image shows geometry translation example with p(3, 0)→p(3 - 8, 0 + 4) and q(6, -6)→q(6 - 8, -6 + 4), and a grid with points p, q, p, q

Explanation:

Step1: Identify the translation vector

From point \( P(3, 0) \) to \( P'(3 - 8, 0 + 4) \), the translation vector is \( (-8, 4) \).

Step2: Apply the translation to point \( Q \)

Let \( Q = (x, y) \). We need to find \( Q' \) using the translation vector \( (-8, 4) \). Wait, but first, we need to know the coordinates of \( Q \). Wait, the problem says "Point Q is the pre - image point". Wait, maybe from the diagram (since the user provided a diagram with Q, but in the text, maybe there was a typo? Wait, looking at the initial part, maybe Q is, for example, if we assume Q's pre - image coordinates (maybe from the grid? Wait, the multiple - choice options are (-2,-2), (2,2), (2,-2), (-2,2). Wait, maybe the pre - image Q has coordinates, let's see the translation: from P(3,0) to P'(3 - 8, 0+4)=(-5,4). Wait, maybe the pre - image Q is, let's suppose the translation is \( (x,y)\to(x - 8,y + 4) \). Let's test the options. Let's assume Q is, say, (6,-6) as in the top part? Wait, no, the multiple - choice is about Q. Wait, maybe the pre - image Q has coordinates, let's solve for Q' given the translation. Wait, maybe the pre - image Q is (6,-6)? No, the multiple - choice options are small. Wait, maybe the pre - image Q is (2, - 2)? Wait, no. Wait, let's use the translation vector. The translation is \( T: (x,y)\to(x - 8,y + 4) \). Wait, no, in the example, P(3,0) becomes P'(3 - 8, 0+4)=(-5,4). Wait, maybe the pre - image Q is (6,-6) becomes Q'(6 - 8,-6 + 4)=(-2,-2). Wait, but the multiple - choice has (-2,-2) as an option. Wait, let's check: if Q is the pre - image, and the translation is \( (x,y)\to(x - 8,y + 4) \), let's suppose Q has coordinates (6,-6), then Q' is (6 - 8,-6 + 4)=(-2,-2). But the multiple - choice options include (-2,-2). Wait, maybe the pre - image Q is (6,-6), but the multiple - choice is about Q's image. Wait, the question is "Point Q is the pre - image point. What are the coordinates of the image, Q'?". So we need to apply the translation to Q. From the example, the translation is \( (x,y)\to(x - 8,y + 4) \). Wait, but we need to know Q's coordinates. Wait, maybe from the grid (the diagram), let's assume Q is (6,-6) (from the top part), then Q' is (6 - 8,-6 + 4)=(-2,-2). But let's check the options. The options are:

• (-2,-2)
• (2,2)
• (2,-2)
• (-2,2)

If we apply the translation \( (x,y)\to(x - 8,y + 4) \) to Q, let's suppose Q is (6,-6), then Q' is (6 - 8, - 6+4)=(-2,-2).

Answer:

A. (-2,-2)