Not every nonlinear function is a polynomial. Quadratics, cubics, and quartics grow forever; rational functions have asymptotes they never touch; irrational functions have a domain that stops short. Compare all of them side by side.
a controls how narrow or wide the parabola is, and whether it opens upward (positive a) or downward (negative a). h and k together give the vertex, the parabola’s highest or lowest point — the turning point highlighted in amber above.
Among the power functions specifically, the exponent’s parity (even or odd) determines end behavior: even-degree functions (quadratic, quartic) always end up pointing the same direction on both sides. Odd-degree functions (cubic) always end up pointing opposite directions — one side rises, the other falls. Recognizing this from a graph alone, without computing a single point, is a skill the SAT tests directly.
Rational and irrational functions break that pattern entirely — they’re not polynomials at all, so “end behavior” gets replaced by asymptotes and domain restrictions instead. Recognizing which kind of boundary a graph is showing you — a turning point, an asymptote, or a hard domain edge — is the real skill this whole topic is building toward.
The SAT usually gives you vertex form directly, asks you to read off the vertex, or gives you the vertex plus one other point and asks you to solve for a — worth practicing both directions.