Question

reflect the figure over the line y = 1/3x + 1. plot all of the points of the reflected figure. you may click a plotted point to delete it.

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 perpendicular line from the point to the line of reflection, finding the intersection point of the perpendicular line and the line of reflection, 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 $m\times m'=-1$). The equation of the perpendicular line passing through $(x_0,y_0)$ is $y - y_0=-3(x - x_0)$.

Step2: Find intersection of lines

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 $y=\frac{1}{3}\times\frac{3(3x_0 + y_0 - 1)}{10}+1=\frac{3x_0 + y_0 - 1}{10}+1=\frac{3x_0 + y_0 - 1 + 10}{10}=\frac{3x_0 + y_0+9}{10}$. 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 $y_i=\frac{y_0 + y_1}{2}$. So $x_1 = 2x_i-x_0$ 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, we get the transformation formula for reflection over $y=\frac{1}{3}x + 1$:
\[

$$\begin{align*} x_1&=\frac{4}{5}x_0+\frac{3}{5}(y_0 - 1)\\ y_1&=\frac{3}{5}(x_0 + 1)+\frac{4}{5}y_0 \end{align*}$$

\]
For each vertex of the given figure (assume vertices are $(x_{vertex},y_{vertex})$), apply the above formulas to find the coordinates of the reflected vertices. Then plot these reflected vertices to get the reflected figure.

Since no specific vertex coordinates of the figure are given in the text, the general method to find the reflected points of a figure over the line $y=\frac{1}{3}x + 1$ is as described above.

Answer:

Use the transformation formulas $x_1=\frac{4}{5}x_0+\frac{3}{5}(y_0 - 1)$ and $y_1=\frac{3}{5}(x_0 + 1)+\frac{4}{5}y_0$ for each vertex $(x_0,y_0)$ of the original figure to find the vertices $(x_1,y_1)$ of the reflected figure and then plot them.