Question

four students are trying to find the rule that translates point \\(n(-2, -4)\\) to \\(n(2, 4)\\). each students reasoning is shown below.

raheem: the rule is \\((x \cdot (-1), y \cdot (-1))\\) because \\(-2 \cdot (-1) = 2\\) and \\(-4 \cdot (-1) = 4\\).
casey: the rule is \\((x + 2, y + 4)\\) because the image is \\((2, 4)\\).
andrew: the rule is \\((x + 4, y + 0)\\) because the coordinates are opposites.
lo: the rule is \\((x + 4, y + 8)\\) because \\(-2 + 4 = 2\\) and \\(-4 + 8 = 4\\).

which student is correct?

raheem
casey
andrew
lo

Explanation:

Identify the given points

We are given a pre-image point \(N(-2, -4)\) and its image point \(N'(2, 4)\) after a translation.

Define translation rule

A translation shifts a point horizontally by \(h\) units and vertically by \(k\) units. The general algebraic rule is:

$$(x, y) ightarrow (x + h, y + k)$$

Calculate horizontal shift

Using the \(x\)-coordinates of \(N\) and \(N'\):

$$-2 + h = 2$$

$$h = 2 - (-2) = 4$$

Calculate vertical shift

Using the \(y\)-coordinates of \(N\) and \(N'\):

$$-4 + k = 4$$

$$k = 4 - (-4) = 8$$

Evaluate student reasoning

The correct translation rule is \((x + 4, y + 8)\). Let's check the students:

• Raheem proposes a dilation/reflection rule: \((x \cdot (-1), y \cdot (-1))\).
• Casey proposes \((x + 2, y + 4)\).
• Andrew proposes \((x + 4, y + 0)\).
• Lo proposes \((x + 4, y + 8)\) because \(-2 + 4 = 2\) and \(-4 + 8 = 4\).

Therefore, Lo is correct.

Answer:

• Raheem
• Casey
• Andrew
• Lo (Correct answer)