Algebra Know-It-ALL

404 Graphs of Quadratic Functions

The absolute maximum is at the point (r, 0). The graph looks as if the parabola “hangs

from” the x axis.

Are you confused?

In Figs. 24-5 and 24-6, you’ll notice horizontal lines with the equation y=k. You might ask, “Why are

the lines there? How do we find the points where the lines intersect the parabolas? Why do we need

the points?” The answer: “Curve construction!” The lines allow us to find points that help us draw

approximations of the graphs, once we’ve found the zero point (r, 0) and have figured out whether

the parabola opens upward or downward. If the parabola opens upward, we should choose a positive

number for k. If the parabola opens downward, we should choose a negative number for k. Then we’ll

know that the line y=k must intersect the parabola at two points. We can find the x-values of those

points by letting y=k in the quadratic function, and then “cooking up” a new quadratic equation out

of that:

ax^2 +bx+c=k

Figure 24-6 Graph of a quadratic function with one

real zero when the coefficient of x^2 is

negative. The parabola opens downward,

is tangent to the x axis at a single

point, and has an absolute maximum

withy= 0. A line with the equation

y=k, where k < 0, intersects the

parabola twice.

x

y

x=q

y=k

x=p

y=k

Liney=k

wherek< 0

x=r

y= 0

Absolute

maximum