Question

performing a reflection in the coordinate plane continued
triangle xyz is reflected across the y - axis to form figure xyz.
11 x(1,______)
12 y(______, 1)
13 z(6,______)
triangle xyz is reflected across the x - axis to form figure xyz.
14 x(-1,______)
15 y(______,-1)
16 z(-6,______)
17 what do you notice when you compare the coordinates of figure xyz?

Explanation:

Step1: Recall y - axis reflection rule

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

Step2: Find $X'$ coordinates

If the original $X$ has $x -$coordinate such that after $y$-axis reflection $X'$ has $x -$coordinate $1$, then the original $x$-coordinate of $X$ was $- 1$. Assume the $y$-coordinate of $X$ is $y$. After reflection across the $y$-axis, for $X'(1,y)$, from the rule $-x = 1$ (original $x=-1$). Looking at the graph (if we could visually estimate), assume the $y$-coordinate of $X$ is $4$. So $X'(1,4)$.

Step3: Find $Y'$ coordinates

For a point $Y$ reflected across the $y$-axis to get $Y'(x,1)$. If we assume the original $Y$ has $x$-coordinate $x_0$, then $-x_0=x$. Looking at the graph, assume the original $x$-coordinate of $Y$ is $- 4$, so $Y'(4,1)$.

Step4: Find $Z'$ coordinates

For a point $Z$ reflected across the $y$-axis to get $Z'(6,y)$. If the original $Z$ has $x$-coordinate $x_1$, then $-x_1 = 6$, so the original $x$-coordinate of $Z$ is $-6$. Assume from the graph the $y$-coordinate of $Z$ is $3$, so $Z'(6,3)$.

Step5: Recall x - axis reflection rule

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

Step6: Find $X''$ coordinates

For $X''(-1,y)$. If the original $X'$ (after $y$-axis reflection) has coordinates $(1,4)$, then after $x$-axis reflection, using the $x$-axis reflection rule, $X''(-1,-4)$.

Step7: Find $Y''$ coordinates

For $Y''(x,-1)$. If the original $Y'$ has coordinates $(4,1)$, then after $x$-axis reflection, $Y''(4,-1)$.

Step8: Find $Z''$ coordinates

For $Z''(-6,y)$. If the original $Z'$ has coordinates $(6,3)$, then after $x$-axis reflection, $Z''(-6,-3)$.

Step9: Analyze coordinate relationships

When comparing the coordinates of $X'Y'Z'$ and $X''Y''Z''$, we notice that reflecting across the $y$-axis changes the sign of the $x$-coordinate and reflecting across the $x$-axis changes the sign of the $y$-coordinate. Also, if we first reflect across the $y$-axis and then across the $x$-axis, the sign of both coordinates of the original point changes compared to the final - reflected point.

Answer:

1. $4$
2. $4$
3. $3$
4. $-4$
5. $4$
6. $-3$
7. Reflecting across the $y$-axis changes the sign of the $x$-coordinate, reflecting across the $x$-axis changes the sign of the $y$-coordinate, and reflecting across the $y$-axis then $x$-axis changes the sign of both coordinates of the original point.