Playing with graphs - (1)

Let $f(x) = y^2 + 2xy + 40|x| - 400$ $\implies f(x) = \begin{cases} y^2 + 2xy + 40x - 400 & \text{for } x \ge 0 \\ y^2 + 2xy - 40x - 400 & \text{for } x < 0 \end{cases}$

For $x \ge 0$ , the value of $y$ as a function of $x$ is given by:

$\begin{aligned} y^2 + 2xy + 40x - 400 & = 0 \\ \implies y & = \frac{-2x \pm \sqrt{4x^2-160x+1600}}{2} \\ & = -x \pm (x - 20) \\ \implies y & = \begin{cases} -20 \\ 20 - 2x \end{cases} \end{aligned}$

Similarly for $x < 0$ :

$\begin{aligned} y^2 + 2xy - 40x - 400 & = 0 \\ \implies y & = \frac{-2x \pm \sqrt{4x^2+160x+1600}}{2} \\ & = -x \pm (x + 20) \\ \implies y & = \begin{cases} 20 \\ -20 - 2x \end{cases} \end{aligned}$

Therefore, $f(x)$ are four lines $\begin{cases} y = \pm 20 \\ y = -2x \pm 20 \end{cases}$ and the bounded area is a parallelogram.

The vertexes of the parallelogram are $\begin{cases} -2x+20 = \pm 20 & \implies \begin{cases} (0,20) \\ (20, -20) \end{cases} \\ -2x-20 = \pm 20 & \implies \begin{cases} (-20,20) \\ (0, -20) \end{cases} \end{cases}$

The area of the parallelogram $= 20 \times 40 = \boxed{800}$

By graphing we get this | |gram points ( - 20,20), (0, 20), (0, - 20), (20, - 20). The base is 20. Height 20 - (- 20)=40.
So the area =20x40=800.