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