Question

what line of reflection will give an image with point d at (-4,5)? x=-3 y=-4 y=x y=-x

Explanation:

Step1: Recall reflection rules

For reflection over the line $y = -x$, if a point $(x,y)$ is reflected, the new - point $(x',y')$ has coordinates $(x',y')=(-y,-x)$.
Let the original point $D$ have coordinates $(x,y)$ and the image $D'$ have coordinates $(-4,5)$.
If we use the rule for reflection over $y=-x$:
If $(x',y') = (-y,-x)=(-4,5)$, then we can solve the system of equations

$$\begin{cases}-y=-4\\-x = 5\end{cases}$$

, which gives

$$\begin{cases}y = 4\\x=-5\end{cases}$$

.
For reflection over the line $y=x$, if a point $(x,y)$ is reflected, the new - point has coordinates $(y,x)$.
For reflection over the line $x = a$ (in this case $a=-3$), the $y$ - coordinate of the point remains the same, and the distance between the $x$ - coordinate of the point and $x = a$ is the same on both sides of the line $x=a$.
For reflection over the line $y = b$ (in this case $b = 4$), the $x$ - coordinate of the point remains the same, and the distance between the $y$ - coordinate of the point and $y = b$ is the same on both sides of the line $y = b$.

Step2: Apply reflection over $y=-x$

Let the original point $D$ have coordinates $(x,y)$. When reflected over $y=-x$, the transformation is $(x,y)\to(-y,-x)$.
If the image is $D'(-4,5)$, then the original point $D$ has coordinates $(-5,4)$.
We can check the other reflection rules:

• Reflection over $x=-3$: Let the point $D$ have coordinates $(x,y)$. The distance between $x$ and $-3$ is $d_1=\vert x + 3\vert$. The $y$ - coordinate remains the same.
• Reflection over $y = 4$: Let the point $D$ have coordinates $(x,y)$. The distance between $y$ and $4$ is $d_2=\vert y - 4\vert$. The $x$ - coordinate remains the same.
• Reflection over $y=x$: If $(x,y)$ is reflected over $y=x$, the new point is $(y,x)$.

The rule for reflection over the line $y=-x$ gives the correct transformation from an unknown original point to the point $D'(-4,5)$.

Answer:

$y=-x$