First Principles AI

Solution

The transformation matrix is:

T=\begin{bmatrix}1&2\\3&4\end{bmatrix}

Let a point on the original curve be $(x,y)$ with:

y=x^2

The image point $(X,Y)$ is:

\begin{bmatrix}X\\Y\end{bmatrix}=T\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}x+2y\\3x+4y\end{bmatrix}

So:

X=x+2x^2,\qquad Y=3x+4x^2

Eliminate $x$ .

From $X=x+2x^2$ and $Y=3x+4x^2$ , compute:

2X-Y=2(x+2x^2)-(3x+4x^2)=-x

Hence:

x=Y-2X

Also:

Y-3X=(3x+4x^2)-3(x+2x^2)=-2x^2 Rightarrow x^2=\frac{3X-Y}{2}

Substitute $x=Y-2X$ into $x^2$ :

(Y-2X)^2=\frac{3X-Y}{2}

Multiply by $2$ :

\boxed{2(Y-2X)^2=3X-Y}

Equivalently:

\boxed{2Y^2-8XY+8X^2+Y-3X=0}