Answer
Intersection points: $(-\displaystyle \frac{\sqrt{3}}{3},-\frac{1}{3})$ and $(\displaystyle \frac{\sqrt{3}}{3},-\frac{1}{3})$
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Work Step by Step
See attached image (desmos.com).
To find the intersection points algebraically,
substitute $y=-x^{2} $ into the second equation:
$-x^{2}=2x^{2}-1$
$0=3x^{2}-1$
$3x^{2}=1$
$x^{2}=\displaystyle \frac{1}{3}$
$x=\displaystyle \pm\sqrt{\frac{1}{3}}=\pm\frac{\sqrt{3}}{3}\approx\pm 0.577$
Back-substitute:
$y=-x^{2}=-\displaystyle \frac{1}{3}$
Intersection points: $(-\displaystyle \frac{\sqrt{3}}{3},-\frac{1}{3})$ and $(\displaystyle \frac{\sqrt{3}}{3},-\frac{1}{3})$
