Algebra Know-It-ALL

Our goal is to find the coordinates of points so we can draw an approximation of the parabola. We can’t get

perfection with pencil and paper, so we don’t need exact values for the coordinates. If we can get within

1/10 of a unit, that ought to be good enough. When we use a calculator to find 801/2, we get roughly

8.944. That’s so close to 9 that it won’t make a difference, when we draw the curve on paper, if we call it

9. So let’s continue with our arithmetic, substituting 9 in for 801/2, as follows:

x= (− 12 ± 9) / (−8)

=−3/(−8) or −21/(−8)

= 3/8 or 21/8

These represent the approximate x-values of the points where the horizontal line intersects the parabola

in Fig. 24-7. The y-values are exactly equal to −5 for both points. We can plot the points as (3/8, −5)

and (21/8, −5). We’ve already determined that (3/2, 0) is on the curve, and that it represents the absolute

maximum. With these three points put down on our graph paper, we can fill in an approximation of the

parabola.

No Real Zeros

When a quadratic equation has no real roots and its associated quadratic function has no

real zeros, its graph is a parabola, but the curve lies entirely on one side of the independent-

variable axis.

Parabola opens upward

Figure 24-8 is a generic graph of a quadratic function of x with no real zeros. The parabola

does not cross the x axis anywhere. The curve has an absolute minimum that lies in the first

or second quadrant. If the quadratic function is

y=ax^2 +bx+c

thena > 0 in this example, because the parabola opens upward.

It’s reasonable to wonder, “Where do we start locating points on the curve in situations

like this?” The task doesn’t look easy at first, but we can find the x-value of the absolute mini-

mum point. If we call that value xmin, then

xmin=−b/(2a)

They-value of the absolute minimum point, ymin, is whatever we get when we plug xmin into

the function:

ymin=axmin^2 +bxmin+c

Once we’ve found the coordinates of the absolute minimum, we can find the coordinates

of two other points. We can pick a number p somewhat smaller than xmin, and we can pick

No Real Zeros 407