Chapter 9.2, Problem 25P

Open-Ended For Exercises 24-26, given an example of a quadratic function with the given characteristic (s).

Its graph lies entirely below the $x-$ axis.

To determine

To find the quadratic function.

Answer to Problem 25P

The required quadratic function is $f\left(x\right)=-x^2-2.$

Explanation of Solution

Given information: Given parent function $y=x^2.$

Calculation:

Use the parent function $y=x^2.$

Which has its vertex at $\left(0,0\right)$ and opens up.

Because the graph lies below the $x-$ axis, then

The parabola opens downward so $a<0$ .

So, retain its coefficient: $a=-1$ for simplicity.

The parent function already opens upward

So, retain its coefficient: $a=1,$ note that $a>0$ in this case.

Then shift the graph down for it to lie below the $x-$ axis, say for function $3$ units down:

Thus, the required quadratic function is $f\left(x\right)=-x^2-2.$