Algebra Know-It-ALL

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.

6. 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