Question

graph the image of the figure using the transformation given. give the coordinates of the image points.

1. reflection across the x - axis
2. reflection across the x - axis
3. reflection across the y - axis

s(-5,0), t(-4,5), u(0,1)

1. reflection across x = 3
2. reflection across y=-1
3. reflection across x = 1

v(-3,-5), u(-3,-2), t(-2,-2), s(2,-5)

Explanation:

Step1: Recall reflection rules

• Reflection across the x - axis: $(x,y)\to(x, - y)$
• Reflection across the y - axis: $(x,y)\to(-x,y)$
• Reflection across the line $x = a$: $(x,y)\to(2a - x,y)$
• Reflection across the line $y = b$: $(x,y)\to(x,2b - y)$

Step2: Solve 1) and 2) (Reflection across x - axis)

For a point $(x,y)$ reflected across the x - axis, the new point is $(x,-y)$. For example, if we have a point $(3,4)$ reflected across the x - axis, it becomes $(3, - 4)$.

Step3: Solve 3) (Reflection across y - axis)

For points $S(-5,0)$, $T(-4,5)$, $U(0,1)$ reflected across the y - axis, we use the rule $(x,y)\to(-x,y)$. So $S'=(5,0)$, $T'=(4,5)$, $U'=(0,1)$.

Step4: Solve 4) (Reflection across $x = 3$)

For a point $(x,y)$ reflected across $x = 3$, the new x - coordinate is $2\times3 - x=6 - x$ and the y - coordinate remains the same.

Step5: Solve 5) (Reflection across $y=-1$)

For a point $(x,y)$ reflected across $y = - 1$, the new y - coordinate is $2\times(-1)-y=-2 - y$ and the x - coordinate remains the same.

Step6: Solve 6) (Reflection across $x = 1$)

For points $V(-3,-5)$, $U(-3,-2)$, $T(-2,-2)$, $S(2,-5)$ reflected across $x = 1$, the new x - coordinate of a point $(x,y)$ is $2\times1 - x=2 - x$ and the y - coordinate remains the same. So $V'=(5,-5)$, $U'=(5,-2)$, $T'=(4,-2)$, $S'=(0,-5)$

Answer:

The new coordinates for each reflection can be found using the above - mentioned rules for each individual point in the given figures. For example, for 3) the new points are $S'(5,0)$, $T'(4,5)$, $U'(0,1)$; for 6) the new points are $V'(5,-5)$, $U'(5,-2)$, $T'(4,-2)$, $S'(0,-5)$ etc.