396CHAPTER24 Graphs of Quadratic Functions
When a quadratic equation is in polynomial standard form but the 0 on the right is
replaced by another variable and then the sides are transposed, we have a quadratic function.
For example,x^2 + 2 x+ 1 = 0is a quadratic equation in x, buty=x^2 + 2 x+ 1is a quadratic function with the independent variable x and the dependent variable y.
The real roots (if any) of a quadratic equation are called zeros when we talk about the
associated quadratic function. The zeros represent the values of the independent variable for
which the dependent variable equals 0.Two Real Zeros
A quadratic equation with real coefficients and a real constant can have two real roots, one real
root, or no real roots. When a quadratic equation has two real roots, its associated quadratic
function has two real zeros. When we graph such a function, it crosses the independent-
variable axis at two distinct points.The parabola
When a quadratic function has real coefficients and a real constant, its graph is a curve called
aparabola. Figure 24-1 shows several such graphs. All parabolas have a characteristic shape
that’s easy to recognize. Some are “narrow” and others are “broad,” but the general contour is
the same in them all. Some “hold water”; they are said to open upward or be concave upward.
Others “spill water”; they are said to open downward or be concave downward. The graph of a
quadratic function always passes the “vertical-line test.”Copyright © 2008 by The McGraw-Hill Companies, Inc. Click here for terms of use.