Question

reflect the figure over the line $y = -\frac{1}{3}x - 1$. plot all of the points of the reflected figure. you may click a plotted point to delete it. answer attempt 1 out of 3

Explanation:

Step1: Recall reflection formula

For a point $(x_0,y_0)$ reflected over the line $y = mx + c$ ($m=-\frac{1}{3}, c = - 1$ here), the general - formula for the reflected point $(x_1,y_1)$ involves finding the intersection of the perpendicular line from the point to the given line and then using the mid - point formula. First, the slope of the perpendicular line to $y=-\frac{1}{3}x - 1$ is $m'=3$ (since the product of slopes of two perpendicular lines is $- 1$). The equation of the perpendicular line passing through $(x_0,y_0)$ is $y - y_0=3(x - x_0)$.

Step2: Find intersection point

We solve the system of equations

$$\begin{cases}y=-\frac{1}{3}x - 1\\y - y_0=3(x - x_0)\end{cases}$$

. Substitute $y$ from the first equation into the second: $-\frac{1}{3}x - 1-y_0=3(x - x_0)$. Expand: $-\frac{1}{3}x - 1-y_0=3x-3x_0$. Rearrange terms: $-\frac{1}{3}x-3x=-3x_0 + y_0 + 1$. Combine like terms: $-\frac{1 + 9}{3}x=-3x_0 + y_0+1$, so $x=\frac{3(3x_0 - y_0 - 1)}{10}$. Then $y=-\frac{1}{3}\times\frac{3(3x_0 - y_0 - 1)}{10}-1=\frac{-3x_0 + y_0 + 1}{10}-1=\frac{-3x_0 + y_0 - 9}{10}$. Let the intersection point be $(x_i,y_i)$.

Step3: Use mid - point formula

Since the intersection point $(x_i,y_i)$ is the mid - point between $(x_0,y_0)$ and $(x_1,y_1)$, we have $x_i=\frac{x_0 + x_1}{2}$ and $y_i=\frac{y_0 + y_1}{2}$. Then $x_1 = 2x_i-x_0$ and $y_1 = 2y_i-y_0$. After substituting $x_i=\frac{3(3x_0 - y_0 - 1)}{10}$ and $y_i=\frac{-3x_0 + y_0 - 9}{10}$ and simplifying, the transformation formulas for reflecting a point $(x,y)$ over the line $y=-\frac{1}{3}x - 1$ are:
\[x'=\frac{4}{5}x-\frac{3}{5}y-\frac{3}{5}\]
\[y'=-\frac{3}{5}x+\frac{4}{5}y-\frac{4}{5}\]
For each vertex of the given polygon, substitute the $x$ and $y$ coordinates into the above formulas to get the coordinates of the reflected vertices. Then plot these new vertices to form the reflected figure.

Since no specific coordinates of the polygon's vertices are given in the text (only the visual graph which we can't directly read coordinates from in this text - based response), the general method for reflection over the line $y =-\frac{1}{3}x - 1$ is shown above. If we assume a vertex of the polygon is $(x_0,y_0)$, we calculate its reflection $(x_1,y_1)$ using the above - derived transformation formulas.

Answer:

Use the transformation formulas $x'=\frac{4}{5}x-\frac{3}{5}y-\frac{3}{5}$ and $y'=-\frac{3}{5}x+\frac{4}{5}y-\frac{4}{5}$ to find the coordinates of the reflected points of the vertices of the figure, and then plot them.