Stanton D. answered • 05/09/20

Tutor to Pique Your Sciences Interest

Hi Cheyanna B.,

I'm a little behind the times on terminology of "slope field", but the intent of the question is clear: for the types of functions (or not!) listed, what are their slopes in the various quadrants. You can eliminate (1) and (2) right off, because they don't **have** portions of the curves which lie in all 4 quadrants (y= x^2, for example, will never have negative y values, will it).

That leaves the circle in (4) or the hyperbola in (3). The circle has the wrong sign of slope in all 4 quadrants, but the hyperbola (as in "Goldilocks and the Three Bears") is j-u-s-t right! So that is the one you want.

Note that for parabolas, hyperbolas, circles, ellipses, and so on displaced from the origin by coordinate changes (y - b = ( x - a )^2 , for example) these regions with certain slope properties will also be displaced.

Now I'll pose you a thought problem. When you consider a parabola of the form y = x^2 , y can never be negative. But that's true only on the real numbers; for complex numbers, y can indeed be negative in that equation, and x is then a complex number (or rather, a pair of them). But to represent a complex number, you need a Cartesian plane, that is, both the original x and y axes. So a function then represents a point-to-point mapping from one Cartesian plane into another Cartesian plane. Could you say that a function has a slope, in that case, and what would that represent?

-- Cheers, -- Mr. d.