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Why does f(x)=-x^2 become a "sad mouth"?

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In this case, it’s -(x^(2)) rather than (-x)^(2).

So we’re taking the negative of x^2 rather than squaring negative x.
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It means f(x)=-(x^2 ). First square and then negate: f(-2)=-(-2)^2 =-4.

Therefore f(x)=-x^2 is always either negative or 0.

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