Chapter 3, Problem 21RE

(a)

To determine

To graph: the given quadratic function

Answer to Problem 21RE

  

Explanation of Solution

Given:

  $f\left(x\right)=\frac{9}{2}{x}^2+3x+1$

Calculation:

For the function $f\left(x\right)=\frac{9}{2}{x}^2+3x+1,a=\frac{9}{2},b=3$ and $c=1$

To find the y-intercept, evaluate the function at $x=0$ , y-intercept: $f\left(0\right)=1$

Now solve $f\left(x\right)=0$ , to find the x-intercepts.

  $\begin{array}{c}\frac{9}{2}{x}^2+3x+1=0\\ x=\frac{-3\pm \sqrt{9-18}}{9}\\ x=\frac{-3\pm \sqrt{-9}}{9}\\ x=\frac{1\pm i}{3}\\ x=\frac{1+i}{3}\text{or}\\ x=\frac{1-i}{3}\end{array}$

No real x-intercept is found. So, the function $f\left(x\right)=\frac{9}{2}{x}^2+3x+1$ has imaginary intercept or no real intercept. Its discriminant $b^2-4ac=9-4\left(\frac{9}{2}\right)\left(1\right)=-9<0$ , so it does not cross or touch the x-axis.

The vertex is at

  $x=-\frac{3}{2\left(\frac{9}{2}\right)}=-\frac{1}{3}$

Since $f\left(-\frac{1}{3}\right)=\frac{1}{2}$ , thus the vertex is $\left(-\frac{1}{3},\frac{1}{2}\right)$ .

  

The graph of $f\left(x\right)=\frac{9}{2}{x}^2+3x+1$ is a parabola that opens up and has its vertex at $\left(-\frac{1}{3},\frac{1}{2}\right)$ .

Its axis of symmetry is the line $x=-\frac{1}{3}$ . Its y-intercept is 1 and it has no real x-intercept.

Conclusion:

Thus, the given equation is drawn.

(b)

To determine

the domain and the range of the function.

Answer to Problem 21RE

The domain is $\left(-\infty ,\infty \right)$ .

The range is $\left(-\infty ,\frac{1}{2}\right)$ .

Explanation of Solution

Calculation:

Determine the range and domain of the function $f\left(x\right)=\frac{9}{2}{x}^2+3x+1$ .

The domain of quadratic equation is all real numbers $\left(-\infty ,\infty \right)$ .

For the range, find the minimum and maximum value of function. The graph of the function $f\left(x\right)=\frac{9}{2}{x}^2+3x+1$ shows that the vertex $\left(-\frac{1}{3},\frac{1}{2}\right)$ of the function is a minimum point.

So, its range is $\left(-\infty ,\frac{1}{2}\right)$ .

Conclusion:

The domain is $\left(-\infty ,\infty \right)$ .

The range is $\left(-\infty ,\frac{1}{2}\right)$ .

(c)

To determine

the increasing and decreasing interval.

Answer to Problem 21RE

Hence, decreasing interval $\left(-\infty ,-\frac{1}{3}\right)$ and increasing interval $\left(-\frac{1}{3},\infty \right)$ .

Explanation of Solution

Calculation:

Determine whether the function $f\left(x\right)=\frac{9}{2}{x}^2+3x+1$ is increasing, decreasing or constant. The graph of the function $f\left(x\right)=\frac{9}{2}{x}^2+3x+1$ shows that the vertex $\left(-\frac{1}{3},\frac{1}{2}\right)$ of thefunction is a minimum point. Hence, decreasing interval $\left(-\infty ,-\frac{1}{3}\right)$ and increasing interval $\left(-\frac{1}{3},\infty \right)$ .

Conclusion:

Hence, decreasing interval $\left(-\infty ,-\frac{1}{3}\right)$ and increasing interval $\left(-\frac{1}{3},\infty \right)$ .