Algebra Know-It-ALL

Parabola opens downward

Figure 24-9 is another generic graph of a quadratic function of x with no real zeros. Again, the
parabola does not cross the x axis. The curve has an absolute maximum in the third or fourth
quadrant. If the function is

y=ax^2 +bx+c

thena < 0 because the parabola opens downward. The x-value of the absolute maximum
point,xmax, is

xmax=−b/(2a)

They-value of the absolute maximum point, ymax, is

ymax=axmax^2 +bxmax+c

Once we’ve found the coordinates of the vertex, we can find the coordinates of two other
points. We can pick a number p smaller than xmax, and we can pick another number q larger
thanxmax. We can then plug p and q into the function for x to find two more points on the
curve. Then the parabola is easy to draw.

Figure 24-9 Graph of a quadratic function with no real

zeros when the coefficient of x^2 is negative.

The parabola opens downward, does not

cross the x axis, has an absolute maximum

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

negative y-value.

x

y

x=p

y=f(p)

x=q

y=f(q)

y 0

xmax= –b/(2a)

<

Absolute maximum

No Real Zeros 409