How should -x^2 be treated in graphing equations?

[The Rev]

I'm graphing equations, and I ran into a snag. I assumed that the graph would be the same for both of the following:

$$y=x^2$$ and $$y=-x^2$$

since any negative number squared is equal to it's absolute value squared.

However, the book showed equation 2 as having an inverted graph of equation 1.

So, I suppose my question is, when I come across $$-x^2$$ should I treat it like $$-(x^2)$$ or like $$(-x)^2$$? IOW, should $$-x$$ be treated as $$-1*x$$ or as a number in and of itself, like $$-2$$?

Thanks.

$$\phi$$

The Rev

 

[whozum]

$$x^2 = (-x)^2$$ by some algebra. However you can show that $$-x^2 \neq x^2$$ by some more algebra!

$$-x^2 = -1 x^2$$

 

[Gale]

the negative sign in -X is just a factor. so when you have $$-x^2$$ you are only squaring the X and not the factor that goes along with it. just like if you had $$2x^2$$ you don't square the two. if you want to square the two, you'd use parenthesis, $$(2x)^2$$ same if you want to square the negative.

as far as how to generally treat -X you do just like i mentioned. you treat the negative as a factor, cause that's all it is. the negative symbol has different meanings, so its best to treat it separately. if you have a negative exponent for example, that's telling you that you've got to flip the fraction. if you have a negative with vectors, that has to do with direction.

another thing to remember is that -X isn't necessarily a negative number. if you plug -2 into that, you get a positive number. so, you aren't just putting a negative sign in front of everything, that negative symbol means you' get the opposite of whatever you put in.

 

[The Rev]

Thanks for the clarification!

$$\phi$$

The Rev