Question

determining coordinates of the pre - image
the sequence of transformations, $r_{o,90^{circ}}circ r_{x - axis}$, is applied to $\triangle xyz$ to produce $\triangle xyz$. if the coordinates of $y$ are $(3,0)$, what are the coordinates of $y?
y( )$

Explanation:

Step1: Reverse the 90 - degree rotation

The rule for a 90 - degree counter - clockwise rotation about the origin $R_{O,90^{\circ}}(x,y)=(-y,x)$. To reverse it, we use the rule for a 90 - degree clockwise rotation $R_{O, - 90^{\circ}}(x,y)=(y, - x)$. Let the coordinates of $Y'$ be $(x_1,y_1)$. After a 90 - degree counter - clockwise rotation about the origin from $Y'$ to $Y''$, if $Y''=(3,0)$, then applying the reverse rotation: $x_1 = 0$ and $y_1=-3$. So the coordinates of $Y'$ are $(0, - 3)$.

Step2: Reverse the x - axis reflection

The rule for a reflection over the x - axis is $r_{x - axis}(x,y)=(x, - y)$. To reverse it, we use the same rule for reflection over the x - axis again. If $Y'=(0, - 3)$ is the result of reflecting $Y$ over the x - axis, then applying the reverse reflection, the coordinates of $Y$ are $(0,3)$.

Answer:

$(0,3)$