has no real roots, so the quadratic function has no real zeros. That means the graph does
not cross the x axis anywhere. If a parabola opens upward and fails to cross the x axis, then
that parabola must lie entirely above the x axis.
- In the polynomial, we have a= 7 and b= 5. The x-value of the absolute minimum
point,xmin, is therefore
xmin=−b/(2a)
=−5 / (2 × 7)
=−5/14
We can find the y-value of the absolute minimum point, ymin, by plugging in xmin to the
function and doing the arithmetic:
ymin= 7 xmin^2 + 5 xmin+ 2
= 7 × (−5/14)^2 + 5 × (−5/14)+ 2
= 7 × 25/196 − 25/14 + 2
= 175/196 − 350/196 + 392/196
=−175/196+ 392/196
= 217/196
= 31/28
The coordinates of the vertex point on the parabola are (−5/14, 31/28).
x
y
Each axis
increment
is 1 unit
(13/8,–121/16)
(1/4,0) (3,0)
Figure C-2 Illustration for the solution to Prob. 4
in Chap. 24.
Chapter 24 683