That is my exact question. If we graph $x^2$ + $y$ $=$ $0$, do we get a downward opening parabola?
Let me explain what actually got me confused. I know that $y$ = $x^2$ is an upward opening parabola because its leading coefficient is positive, or in other words, its double derivative is positive.
Likewise, $y$ $=$ $-x^2$ is a downward opening parabola because its double derivative is negative.
We can rewrite $y$ = $-x^2$ as $y$ + $x^2$ $=$ $0$.
Now, this equation $y$ + $x^2$ = $0$ should be a downward opening parabola too, because its basically the same equation as $y$ = $-x^2$.
But I’m not convinced that it is indeed a downward opening parabola. How do I convince myself?
Best Answer
Perhaps noting that since $x^2\ge0$, $y+x^2=0$ implies that $y$ is not positive.