Chapter 8.3, Problem 13Q

To determine

To graph the function:

  $f\left(x\right)=-4x^2-4x+7$

And also define it’s domain and range.

Answer to Problem 13Q

  

Domain is: $x\in \left(-\infty ,\infty \right)$ , and range is: $y\le 8$ .

Explanation of Solution

Given information:

  $f\left(x\right)=-4x^2-4x+7$

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:

  $f\left(x\right)=-4x^2-4x+7$

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,

  $a=-4,b=-4,\text{ and }c=7$

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

  $\begin{array}{c}x=-\frac{b}{2a}\\ =-\frac{-4}{2\left(-4\right)}\\ =-\frac{1}{2}\\ \Rightarrow x=-\frac{1}{2}\end{array}$

Now, find x -coordinate of the vertex of graph of this function, that is $-\frac{b}{2a}=-\frac{1}{2}$

And y -coordinate of vertex is found by putting $x=-\frac{1}{2}$ in equation of function, as:

  $\begin{array}{c}f\left(-\frac{1}{2}\right)=-4{\left(-\frac{1}{2}\right)}^2-4\left(-\frac{1}{2}\right)+7\\ =-4\times \frac{1}{4}+2+7\\ =-1+9\\ =8\end{array}$

Thus, vertex of graph of given function is: $\left(-\frac{1}{2},8\right)$ .

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, $c=7$ . Thus, y -intercept is $\left(0,-4\right)$ .

Since, $a=-4<0$ , thus curve of this is open downward.

From the above information plot the vertex, $\left(-\frac{1}{2},8\right)$ , and y -intercept $\left(0,7\right)$ and make a smooth curve through these points that is open downward and use the axis of symmetry $x=-\frac{1}{2}$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 less than and equals to 8, thus range is $y\le 8$ .