Question

if the figure below were reflected across the y-axis, where would the points p, q, and r be? graph p select q select r select

Explanation:

Step1: Find original coordinates

First, identify the original coordinates of points \( P \), \( Q \), and \( R \) from the graph.

• Point \( P \): Looking at the graph, \( P \) is at \( (3, 4) \).
• Point \( Q \): \( Q \) is at \( (-2, 1) \).
• Point \( R \): \( R \) is at \( (1, -5) \) (Wait, no, looking again, the graph: R is at (1, -5)? Wait, no, the x-axis and y-axis: let's recheck. Wait, the grid: x from -5 to 5, y from -5 to 5. Let's see:

Wait, the original points:

• \( P \): x=3, y=4, so \( (3, 4) \)
• \( Q \): x=-2, y=1, so \( (-2, 1) \)
• \( R \): x=1, y=-5? Wait, no, the point R is at (1, -5)? Wait, the graph shows R at (1, -5)? Wait, no, looking at the y-axis, the bottom is -5, and R is at x=1, y=-5? Wait, maybe I misread. Wait, the reflection over y-axis: the rule is \( (x, y) \to (-x, y) \).

Step2: Apply reflection rule

The rule for reflecting a point \( (x, y) \) across the \( y \)-axis is \( (x, y) \to (-x, y) \).

• For \( P(3, 4) \): Apply the rule: \( x = 3 \to -3 \), \( y \) remains 4. So \( P'(-3, 4) \).
• For \( Q(-2, 1) \): Apply the rule: \( x = -2 \to 2 \), \( y \) remains 1. So \( Q'(2, 1) \).
• For \( R(1, -5) \): Wait, no, wait the original R: looking at the graph, R is at (1, -5)? Wait, no, the graph: the point R is at (1, -5)? Wait, maybe I made a mistake. Wait, the x-coordinate of R: in the graph, R is at x=1, y=-5? Wait, no, let's check again. Wait, the grid: the x-axis has 0, 1, 2, 3, 4, 5 on the right, -1, -2, -3, -4, -5 on the left. The y-axis has 0, 1, 2, 3, 4, 5 on top, -1, -2, -3, -4, -5 on bottom. So:

• \( P \): x=3, y=4 (since it's 3 units right on x, 4 up on y)
• \( Q \): x=-2, y=1 (2 units left on x, 1 up on y)
• \( R \): x=1, y=-5? Wait, no, the point R is at (1, -5)? Wait, the graph shows R at (1, -5)? Wait, maybe the original R is (1, -5)? Wait, no, looking at the graph, the point R is at (1, -5)? Wait, the vertical line at x=1, horizontal at y=-5. So R is (1, -5). Then reflecting over y-axis: \( (1, -5) \to (-1, -5) \). Wait, but maybe I misread R's coordinates. Wait, maybe R is at (1, -5)? Let's confirm:

Wait, the problem says "the figure below were reflected across the y-axis". Let's re-express the original coordinates:

• \( P \): (3, 4)
• \( Q \): (-2, 1)
• \( R \): (1, -5) [Wait, no, maybe R is at (1, -5)? Wait, the graph: the point R is at (1, -5)? Let's check the reflection rule again.

Reflection over y-axis: \( (x, y) \mapsto (-x, y) \).

So:

• \( P(3, 4) \) reflected: \( (-3, 4) \)
• \( Q(-2, 1) \) reflected: \( (2, 1) \)
• \( R(1, -5) \) reflected: \( (-1, -5) \)

Wait, but maybe I made a mistake in R's coordinates. Let's check the graph again. The point R is at (1, -5)? Wait, the x-coordinate is 1 (right of y-axis), y-coordinate is -5 (below x-axis). So yes. Then:

• \( P' \): (-3, 4)
• \( Q' \): (2, 1)
• \( R' \): (-1, -5)

Answer:

• \( P' \): \((-3, 4)\)
• \( Q' \): \((2, 1)\)
• \( R' \): \((-1, -5)\)