Algebra Know-It-ALL

398 Graphs of Quadratic Functions

Whenever a parabola opens upward, it has a single point at which it “bottoms out.” This

point is called the absolute minimum of the graph. In a few moments, we’ll discover how this

point can be located.

Parabola opens downward

Figure 24-3 shows a generic graph of a quadratic function of x with two real zeros, again

calledr and s. But this parabola opens downward. If r is the smaller of the two zeros and s is

the larger, then

r= [−b+ (b^2 − 4 ac)1/2] / (2a)

and

s= [−b− (b^2 − 4 ac)1/2] / (2a)

Note the subtle difference between these two equations and those for the case where the

parabola opens upward! Because of this, the values of r and s are reversed, as compared to their

x

y

x=r x=s

y= 0 y= 0

y< 0

Absolute minimum

xmin= (r+s)/ 2

Figure 24-2 Graph of a quadratic function with

two real zeros when the coefficient

ofx^2 is positive. The parabola opens

upward, crosses the x axis twice, and

has an absolute minimum with y < 0.