# How to do this question 10?

Then teach the underlying concepts
Don't copy without citing sources
preview
?

#### Explanation

Explain in detail...

#### Explanation:

I want someone to double check my answer

1
Feb 12, 2018

$\textcolor{b l u e}{y = - x - 12}$

#### Explanation:

All points on the line $y = - 2 x + 8$ will be transformed by the matrix $\boldsymbol{A}$

Any points on this line will have coordinates of the form $\left(k , - 2 k + 8\right)$

$\therefore$

$\left(\begin{matrix}x ' \\ y '\end{matrix}\right) = \left(\begin{matrix}0 & - 2 \\ - 4 & 0\end{matrix}\right) \left[\left(\begin{matrix}k \\ - 2 k + 8\end{matrix}\right) + \left(\begin{matrix}- 2 \\ 2\end{matrix}\right)\right]$

$\textcolor{w h i t e}{888888} = \left(\begin{matrix}0 & - 2 \\ - 4 & 0\end{matrix}\right) \left(\begin{matrix}k - 2 \\ - 2 k + 10\end{matrix}\right)$

$\textcolor{w h i t e}{888888} = \left(\begin{matrix}4 k - 20 \\ - 4 k + 8\end{matrix}\right)$

i.e.

$x ' = 4 k - 20$ and $y ' = - 4 k + 8$

Eliminating $k$:

$k = \frac{x + 20}{4}$

$y = - 4 \left(\frac{x + 20}{4}\right) + 8$

$\textcolor{b l u e}{y = - x - 12}$

So all images of $\left(x ' , y '\right)$ lie on the line $y = - x - 12$.

The line $y = - x - 12$ is the image of $y = - 2 x + 8$ under the transformation.

Note:

We could have found this line and alternate way. If we had generated two pairs of coordinates using $\boldsymbol{y = - 2 x + 8}$, and used them in the transformation i.e for $\boldsymbol{X}$, we would have 2 pairs of coordinates of the image. This would enable us to find the equation of the line.

Example:

From $y = - 2 x + 8$

For $x = 3$ and $x = 4$

$y = 2$ and $y = 0 \textcolor{w h i t e}{88}$ respectively.

Under the transformation we have:

$x = - 8$ and $y = - 4$

$x = - 4$ and $y = - 8$

$\frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{- 8 - \left(- 4\right)}{- 4 - \left(- 8\right)} = - 1$

$y - \left(- 4\right) = - \left(x - \left(- 8\right)\right)$

$y + 4 = - x - 8$

$y = - x - 12 \textcolor{w h i t e}{888}$ as expected.

• 7 minutes ago
• 8 minutes ago
• 10 minutes ago
• 12 minutes ago
• A minute ago
• 2 minutes ago
• 3 minutes ago
• 3 minutes ago
• 6 minutes ago
• 6 minutes ago
• 7 minutes ago
• 8 minutes ago
• 10 minutes ago
• 12 minutes ago