Question

if the figure below has been reflected across the x-axis, what are the coordinates of the preimage? p
q
r

Explanation:

Step1: Recall reflection over x - axis rule

The rule for reflecting a point \((x,y)\) across the \(x\) - axis is \((x,y)\to(x, - y)\). This means that to find the pre - image from the image (after reflection), if the image point is \((x,y)\), the pre - image point \((x_{pre},y_{pre})\) satisfies \(x_{pre}=x\) and \(y_{pre}=-y\) (or we can say that if the image is \((x,y)\) after reflection over \(x\) - axis, the pre - image is \((x, - y)\) before reflection? Wait, no. Wait, if we have a pre - image \(P(x,y)\) and we reflect it over the \(x\) - axis to get \(P'(x, - y)\). So to find the pre - image from the image, if \(P'\) has coordinates \((x,y)\), then \(P\) (pre - image) has coordinates \((x, - y)\).

Step2: Find coordinates of \(P'\), \(Q'\), \(R'\) from the graph

• From the graph, the coordinates of \(P'\) are \((3,4)\). Using the reflection rule (to find pre - image \(P\) from image \(P'\)), since \(P'\) is the image after reflecting \(P\) over \(x\) - axis, if \(P=(x,y)\), then \(P'=(x, - y)\). So if \(P'=(3,4)\), then \(P=(3, - 4)\) (because \(y_{pre}=-y_{image}\), here \(y_{image} = 4\), so \(y_{pre}=-4\) and \(x_{pre}=x_{image}=3\)).
• The coordinates of \(Q'\) are \((-2,1)\). Using the reflection rule, for pre - image \(Q\), \(x=-2\) (same as \(x\) of \(Q'\)) and \(y=-1\) (since \(y_{image}=1\), \(y_{pre}=-1\)). So \(Q=(-2, - 1)\).
• The coordinates of \(R'\) are \((2, - 5)\). Using the reflection rule, for pre - image \(R\), \(x = 2\) (same as \(x\) of \(R'\)) and \(y = 5\) (since \(y_{image}=-5\), \(y_{pre}=-(-5)=5\)). So \(R=(2,5)\).

Answer:

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