Question

4 noor draws △pqr in the coordinate plane as shown. then she reflects △pqr across the y-axis to form △p′q′r′. doug incorrectly says the coordinates of q′ are (-3, 2). what are the correct coordinates? how might doug have gotten the coordinates (-3, 2)? 5 the points given below are the vertices of △vwx and its image after a reflection. v(2, 1), w(3, 5), x(4, 2) v′(-2, 1), w′(-3, 5), x′(-4, 2) a. what do you notice when you compare corresponding x-coordinates and corresponding y-coordinates of the vertices of △vwx and △v′w′x′? b. what is the line of reflection?

Explanation:

Problem 4

Step1: Find original Q coordinates

From the graph, Q is at \((-3, -2)\)? Wait, no, looking at the grid: Q is at \((-3, -2)\)? Wait, the y-axis: the grid has y from -6 to 4. Wait, the original Q: let's see, the x-coordinate of Q is -3 (since it's 3 units left of y-axis), y-coordinate: looking at the grid, Q is at y = -2? Wait, no, the graph shows Q above P and R. Wait, the grid: x from -6 to 6, y from -6 to 4. Q is at (-3, -2)? Wait, no, maybe I misread. Wait, the problem says Doug says Q' is (-3,2). Wait, reflection over y-axis: rule is \((x,y) \to (-x,y)\). So first, find original Q. Let's look at the graph: Q is at (3, -2)? Wait, no, the grid: the x-axis is horizontal, y-axis vertical. The triangle is in the third quadrant? Wait, P is at (-5, -4), Q at (-3, -2), R at (-1, -5)? Wait, no, the graph: Q is at (-3, -2)? Wait, no, the y-axis: the numbers on y-axis are 4, 2, 0, -2, -4, -6. So Q is at x = -3, y = -2? Wait, no, the problem says Doug says Q' is (-3,2). Wait, reflection over y-axis: if original Q is (3, -2), then reflection would be (-3, -2). But Doug says (-3,2). So maybe original Q is (3, -2)? Wait, no, let's re-examine.

Wait, the graph: Q is at (-3, -2)? Wait, the x-coordinate: left of y-axis is negative, right is positive. So Q is at x = -3, y = -2? Then reflection over y-axis: (x,y) → (-x, y). So (-3, -2) → (3, -2). But Doug says (-3,2). So maybe Doug reflected over x-axis? Because reflection over x-axis is (x,y) → (x, -y). So if original Q is (-3, -2), reflection over x-axis is (-3, 2), which is what Doug did. So correct reflection over y-axis: original Q is (-3, -2)? Wait, no, maybe I got the original coordinates wrong. Wait, the grid: let's count the squares. From the y-axis (x=0), moving left 3 units: x=-3. For y: between -2 and -4? Wait, the Q is at y = -2? Wait, the problem's graph: Q is at (-3, -2)? Then reflection over y-axis: (x,y) → (-x, y) → (3, -2). But Doug says (-3,2). So Doug probably reflected over x-axis instead of y-axis. So correct coordinates of Q' (reflection over y-axis) would be (3, -2). Wait, no, maybe original Q is (3, -2)? Wait, maybe I mixed up x and y. Wait, the problem says "reflects △PQR across the y-axis". So reflection over y-axis: (x, y) → (-x, y). So first, find original Q's coordinates. From the graph, Q is at (3, -2)? Wait, no, the x-axis: positive to the right, negative to the left. So if Q is at x = -3 (left of y-axis), y = -2 (below x-axis). Then reflection over y-axis: (-3, -2) → (3, -2). Doug says (-3, 2), which is reflection over x-axis: (-3, -2) → (-3, 2). So correct coordinates of Q' are (3, -2). Doug might have reflected over the x - axis instead of the y - axis.

Step2: Confirm reflection rule

Reflection over y - axis: \((x,y)\to(-x,y)\). If original Q is \((-3, -2)\), then Q' should be \((3, -2)\). Doug used reflection over x - axis: \((x,y)\to(x, -y)\), so \((-3, -2)\to(-3, 2)\).

Step1: Compare x - coordinates

For \(V(2,1)\) and \(V'(-2,1)\): \(2\) and \(-2\) are opposites (sum to 0). For \(W(3,5)\) and \(W'(-3,5)\): \(3\) and \(-3\) are opposites. For \(X(4,2)\) and \(X'(-4,2)\): \(4\) and \(-4\) are opposites.

Step2: Compare y - coordinates

For \(V\) and \(V'\): \(1 = 1\). For \(W\) and \(W'\): \(5 = 5\). For \(X\) and \(X'\): \(2 = 2\). So corresponding \(y\) - coordinates are equal, and corresponding \(x\) - coordinates are opposites (i.e., \(x'=-x\) and \(y' = y\)).

Step1: Recall reflection rules

Reflection over \(y\) - axis: \((x,y)\to(-x,y)\), which matches the pattern we saw ( \(x\) - coordinates are negated, \(y\) - coordinates stay same). Reflection over \(x\) - axis: \((x,y)\to(x, -y)\), which doesn't match here. Reflection over \(y = x\): \((x,y)\to(y,x)\), which also doesn't match. So the line of reflection is the \(y\) - axis ( \(x = 0\) ).

Answer:

Correct coordinates of \(Q'\) are \((3, -2)\). Doug might have reflected \(\triangle PQR\) across the \(x\) - axis instead of the \(y\) - axis.

Problem 5a