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:

[tex]y=x^2[/tex] and [tex]y=-x^2[/tex]

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 [tex]-x^2[/tex] should I treat it like [tex]-(x^2)[/tex] or like [tex](-x)^2[/tex]? IOW, should [tex]-x[/tex] be treated as [tex] -1*x[/tex] or as a number in and of itself, like [tex]-2[/tex]?

Thanks.

[tex]\phi[/tex]

The Rev

 

[whozum]

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

[tex] -x^2 = -1 x^2 [/tex]

 

[Gale]

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

[tex]\phi[/tex]

The Rev