Question

what line of reflection will give an image with point f at (1,-5)? y=x y=0 y=-x x=0

Explanation:

Step1: Recall reflection rules

Reflection across $y = x$ swaps $x$ and $y$ - coordinates. Reflection across $y = 0$ (x - axis) changes the sign of the $y$ - coordinate. Reflection across $y=-x$ swaps and negates the coordinates. Reflection across $x = 0$ (y - axis) changes the sign of the $x$ - coordinate.

Step2: Analyze each option

Let's assume the original point $F$ has coordinates $(x,y)$.

• For reflection across $y = x$, the image point $F'$ has coordinates $(y,x)$.
• For reflection across $y = 0$, if $F=(x,y)$ then $F'=(x, - y)$.
• For reflection across $y=-x$, if $F=(x,y)$ then $F'=(-y,-x)$.
• For reflection across $x = 0$, if $F=(x,y)$ then $F'=(-x,y)$.

Step3: Determine the correct reflection

We want to get from an original point $F$ to $F'=(1,-5)$. If we assume the original point $F$ is $(- 5,1)$ and reflect it across $y=-x$, when we swap and negate the coordinates of $(-5,1)$: $x=-5$ and $y = 1$, then the image point has $x$ - coordinate $-y=-1$ and $y$ - coordinate $-x = 5$ which is wrong. If we reflect across $y = 0$, we can't get from a general point to $(1,-5)$ in this way. If we reflect across $x = 0$, we also can't get to $(1,-5)$. If we assume the original point $F$ is $(-5,1)$ and reflect it across $y = x$, we swap the coordinates and get $(1,-5)$.

Answer:

$y=x$