# How do you find the end behavior of a quadratic function?

Sep 13, 2014

Quadratic functions have graphs called parabolas.  The first graph of y = ${x}^{2}$ has both "ends" of the graph pointing upward. You would describe this as heading toward infinity. The lead coefficient (multiplier on the ${x}^{2}$) is a positive number, which causes the parabola to open upward.

Compare this behavior to that of the second graph, f(x) = $- {x}^{2}$.
Both ends of this function point downward to negative infinity. The lead coefficient is negative this time.

Now, whenever you see a quadratic function with lead coefficient positive, you can predict its end behavior as both ends up. You can write: as $x \to \setminus \infty , y \to \setminus \infty$ to describe the right end, and
as $x \to - \setminus \infty , y \to \setminus \infty$ to describe the left end.

Last example: Its end behavior:
as $x \to \setminus \infty , y \to - \setminus \infty$ and as $x \to - \setminus \infty , y \to - \setminus \infty$
(right end down, left end down)