Moving origin to (m,n)

L2MAEL

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Moving the origin to the point $(m,n)$ is also moving the graph, since moving the origin on Desmos is not possible, and the exam requires you to explain this.

Explaining:

Imagine this: “A U-shaped quadratic graph’s vertex is on the center of the + shaped line representing the $x$ and $y$ axis. Now, imagine that the graph kept its position, but the + shaped line representing the $x$ and $y$ axis moved to the left. Is the graph now placed on the left, or right side of the $y$-axis?”

Visual example:

(normally that explains all the concept of this origin movement)

The answer is ‘right side’.

When you move the origin to the left by 2, the vertex of the graph appears at $(2,0)$ in the new coordinate system, because moving the origin shifts the graph in the opposite direction of the origin’s movement.

For example, to move the origin of the graph to $(m,n)$:

$y=x^2$

I would have to change $x$ to '$x+m$', and $y$ to '$y+n$'.

The equation would be:

$y+n=(x+m{)}^2$

$y=(x+m{)}^2-n$

If a domain/range exists, you must replace the $x⁄y$ to the repacement as well.

Applying:

Here’s our graph (visual can be found here: https://www.desmos.com/calculator/hwiengmumv)

$y=0.01\left(x+5\right)\left(x+0\right)\left(x-5\right)\left\{-5\le x\le 10\right\}$

To move the origin to $(m,n)$, we can replace the $x$ and $y$:

$y+n=0.01\left(x+m+5\right)\left(x+m+0\right)\left(x+m-5\right)\left\{-5\le x+m\le 10\right\}$

$y=0.01\left(x+m+5\right)\left(x+m+0\right)\left(x+m-5\right)-n\left\{-5\le x+m\le 10\right\}$