Quadratic functions

A quadratic function f may be written in either of two forms:

You can move from the second form to the first by expanding; you can move from the first to the second by completing the square or by using these formulas:

(We may assume that a ≠ 0, because otherwise our quadratic function is simply a linear function, which we already know how to handle.)

If (as we assume) it's not linear, then the graph of a quadratic function is a shape called a parabola. The point (h, k) on the graph is called the vertex of the parabola. This vertex is the only turning point of the graph; the graph turns upwards at the vertex is if a > 0, and it turns downwards if a < 0. The domain of any quadratic function is the set of all real numbers, while the range is [k, ∞) if a > 0 or (−∞, k] if a < 0. The parabola is symmetric, with a vertical line of symmetry whose equation is x = h. Since f(0) = c, the vertical intercept, or y-intercept, on the graph is (0, c). The horizontal intercepts, or x-intercepts, are (r+, 0) and (r, 0), where r+ and r are given by the quadratic formula:

(It doesn't really matter which one is r+ and which one is r.) However, these will be imaginary numbers if b2 − 4ac is negative, or equivalently if k has the same sign as a, in which case the graph has no horizontal intercepts.

In general, there are up to 7 useful points on the graph:

Some of these points might happen to be the same as others, and the last two won't exist on the graph if the roots r± are imaginary. However, there are always at least three distinct real points on this list.
Go back to the course homepage.
This web page was written by Toby Bartels, last edited on 2019 December 3. Toby reserves no legal rights to it.

The permanent URI of this web page is http://tobybartels.name/MATH-1100/2019FA/parabolas/.