Algebra Know-It-ALL

408 Graphs of Quadratic Functions

another number q somewhat larger than xmin. We can choose integers for these numbers to

make the arithmetic easy. The numbers p and q don’t have to be equally smaller and larger

thanxmin, although we should try to get close to that ideal. Once we’ve chosen the numbers

p and q, we can plug them into the function for x and find two more points on the curve.

Then the parabola is easy to draw.

Are you astute?

Do you wonder why we didn’t use the “x-value first” point-finding strategy earlier in this chapter for

quadratic functions with one or two real zeros? “Isn’t it easier,” you might ask, “to choose a clean integer

x-value for a point, and figure the y-value by plugging into the function? Isn’t that better than picking a

y-value and then grinding through the quadratic formula to get two x-values that are likely to come out

irrational?” You tell me! By choosing x-values first, we have to go through the arithmetic twice, but it’s

usually simple. By choosing the y-value first, the arithmetic can be a little rough, but we only have to do it

once. It’s your choice. Either method will work fine.

Figure 24-8 Graph of a quadratic function with no real

zeros when the coefficient of x^2 is positive.

The parabola opens upward, does not

cross the x axis, has an absolute minimum

with an x-value equal to −b/(2a), and has a

positive y-value.

x

y

y> 0

Absolute minimum

xmin=–b/(2a)

x=p

y=f(p)

x=q

y=f(q)