# patterns for u-shaped graphs

When the equation is $Ax + By = C$, you know it will be a straight line. Is there a specific pattern to know (without plotting $x$ and $y$ yet) that the graph will be u-shaped? For example, the equation $y = x^2 - 9x – 12$ forms a u-shape. But how would you know that by looking at it? How would you know that, for example, it's not a L shape or something else? Is there an equation, as there is in the straight line?

Thanks for any response.

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Seems a little more subjective than you might think. Can I ask: what is it about $Ax+By+C$ that makes it obvious to you that it's a line? Aside from seeing it graphed, of course. Parabolas will always be a (possibly inverted) U shape, though there are other functions that will do so. Also, how much calculus do you know? – Robert Mastragostino Jul 15 '12 at 2:36
This is strictly algebraic question. Not calculus. With t Ax+By+C, no matter what I substitute for x and y, it always creates straight line. – JohnMerlino Jul 15 '12 at 2:40

In addition to Alex' excellent answer, I'd like to contribute the following which may be more accessible if you haven't heard about limits before and gives a slightly more concrete criterion for the 'u-shapes'. I will also assume that by u-shape you mean things that actually look a bit like a 'u' rather than just becoming large on both sides.

The u-shape you describe is called a parabola.

And indeed, you can recognize many of these by their equation:

First look at the graph of $y=x^2$, the simplest example of such a parabola. Now, if you have any equation like $y=Ax^2+Bx+C,\ A>0$, you can complete the square:

$Ax^2+Bx+C=A(x^2+\frac{B}{A}x+\frac{B^2}{4A^2}+\frac{C}{A}-\frac{B^2}{4A^2})=A(x+\frac{B}{2A})^2 + C-\frac{B^2}{4A}$, so this is the simple parabola you saw before, moved to the left by a distance of $\frac{B}{2A}$, stretched in $y$-direction by a factor of $A$ and finally moved upwards by a distance of $C-\frac{B^2}{4A}$.

If $A<0$, your parabola is turned upside down.

For equations with higher powers of $x$, it is more complicated to find out what its graph looks like. As Alex said, odd degrees (highest powers) never give u-shapes, while even degrees can give u-shapes but also 'w-shapes' - consider for example $x^4-3x^2+1$ and more intricate shapes.

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I think all the answers are good. – JohnMerlino Jul 16 '12 at 0:19

If I understand your question, you want to know when a function $f(x)$ will become arbitrarily large as $x$ moves away from some fixed point. Formally, this can be phrased as $$\lim\limits_{x\to +\infty}f(x)=\lim\limits_{x\to -\infty}f(x)=+\infty$$ which in general is hard to test, but is easy for a polynomial. If a polynomial has even degree, then the limits at $\pm \infty$ are the same, while if it is of odd degree they are different. Thus we want polynomials of even degree. And if a polynomial of even degree has positive leading coefficient, both these limits will be $+\infty$ while if it has negative leading coefficient they will be $-\infty$.

Thus the polynomials with "u-shaped" graphs are those of even degree with positive leading coefficient.

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The idea is essentially this:

1 There should only be one minimum

2 It should be purely increasing as you go farther from the minimum

and possibly

3 The rate at which it increases can't slow down (no "waviness" to it)

The first two can be satisfied here quite nicely by completing the square, as us2012 said. However the answer in general (mostly because of the third point) essentially has to involve calculus. The "rate of the rate of increase" (which determines the waviness, or curvature) is very difficult to reason about without it. If you don't care about the third point, convex functions seem to be exactly what you're talking about.

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