Chapter 8.3, Problem 13E

To determine

To graph the function:

  $f\left(x\right)=2x^2+12x+4$

And also define it’s domain and range.

Answer to Problem 13E

  

Domain is: $x\in \left(-\infty ,\infty \right)$ , and range is: $y\ge -14$ .

Explanation of Solution

Given information:

  $f\left(x\right)=2x^2+12x+4$

Concept Used:

Graph the function $f\left(x\right)=a{x}^2+bx+c$

• The graph opens up when $a>0$ , and the graph opens down when $a<0$ .
• The y -intercept is c.
• The x -coordinate of the vertex of graph is $-\frac{b}{2a}$ .
• The axis of symmetry is $x=-\frac{b}{2a}$ .

Calculation:

In order to graph the function:

  $f\left(x\right)=2x^2+12x+4$

First find the axis/line of symmetry of graph. Axis of symmetry of a function $f\left(x\right)=a{x}^2+bx+c$ is given by:

  $x=-\frac{b}{2a}$

Here,

  $a=2,b=12,\text{ and }c=4$

Thus, axis of symmetry of graph of given function is:

  $\begin{array}{c}x=-\frac{b}{2a}\\ =-\frac{12}{2\left(2\right)}\\ =-\frac{12}{4}\\ =-3\\ \Rightarrow x=-3\end{array}$

Now, find x -coordinate of the vertex of graph of this function, that is

  $-\frac{b}{2a}=-3$

And y -coordinate of vertex is found by putting $x=-3$ in equation of function, as:

  $\begin{array}{c}f\left(-3\right)=2{\left(-3\right)}^2+12\left(-3\right)+4\\ =2\times 9-36+4\\ =18-32\\ =-14\end{array}$

Thus, vertex of graph of given function is: $\left(-3,-14\right)$ .

Now, find y-intercept to find two more points on the graph. Since y -intercept of the function $f\left(x\right)=a{x}^2+bx+c$ is given by c .

And, $c=4$ . Thus, y -intercept is $4$ .

Now, since $a=2>0$ , thus curve of this is open upward.

From the above information plot the vertex, $\left(-3,-14\right)$ , and y -intercept $\left(0,4\right)$ and make a smoth curve through these points that is open upward and use the axis of symmetry $x=-3$ to draw other half of the graph. Using a graphing utility this can be represented as:

  

Now, from the above graph it can be concluded that the domain of function is real number, that is $x\in \left(-\infty ,\infty \right)$ , but the range is every number strictly greater than equals to $-14$ , thus range is

  $y\ge -14$ .