Chapter 3, Problem 6CT

(a)

To determine

whether the graph opens up or down or not

Answer to Problem 6CT

The parabola opens up.

Explanation of Solution

Given:

  $f\left(x\right)=3x^2-12x+4$

Calculation:

Compare $f\left(x\right)=3x^2-12x+4$ to $f\left(x\right)=a{x}^2+bx+c$ and find the values $a=3,b=-12$ and $c=4$ . Since $a=3>0$ , the parabola opens up.

Conclusion:

The parabola opens up.

(b)

To determine

the vertex.

Answer to Problem 6CT

Hence the vertex is $\left(2,-8\right)$ .

Explanation of Solution

Calculation:

Now find vertex $\left(-\frac{b}{2a},f\left(\frac{-b}{2a}\right)\right)$

The x-coordinate of the vertex is

  $\begin{array}{c}x=\frac{-b}{2a}\\ =-\frac{-12}{2\left(3\right)}\\ =\frac{12}{6}\\ =2\end{array}$

And the y-coordinate of the vertex is

  $\begin{array}{c}f\left(-\frac{b}{2a}\right)=f\left(2\right)\\ =3{\left(2\right)}^2-12\left(2\right)+4\\ =12-24+4\\ =-8\end{array}$

Conclusion:

Hence the vertex is $\left(2,-8\right)$ .

(c)

To determine

the axis of symmetry.

Answer to Problem 6CT

Therefore the axis of symmetry is $x=2$ .

Explanation of Solution

Calculation:

The axis of symmetry is the line

  $\begin{array}{c}x=-\frac{b}{2a}\\ =2\end{array}$

Conclusion:

Therefore the axis of symmetry is $x=2$ .

(d)

To determine

the intercepts.

Answer to Problem 6CT

Hence the x intercepts is $\left(\frac{6+2\sqrt{6}}{3},0\right)$ and $\left(\frac{6-2\sqrt{6}}{3},0\right)$ .

Explanation of Solution

Calculation:

Now find the y-intercept, $f\left(0\right)$

  $\begin{array}{c}f\left(0\right)=3{\left(0\right)}^2-12\left(0\right)+4\\ =4\end{array}$

Hence the y intercepts is $\left(0,4\right)$ .

Find the discriminant, $b^2-4ac$ , to determine, the number of the x-intercepts if any.

  $\begin{array}{c}{b}^2-4ac={\left(-12\right)}^2-4\cdot 4\cdot 3\\ =144-48\\ =96\end{array}$

Since the value of the discriminant is positive, the parabola will have two different x-intercepts.

Now find the x-intercepts by solving $f\left(x\right)=0$ ,

  $\begin{array}{c}3x^2-12x+4=0\\ x=\frac{12\pm \sqrt{144-48}}{6}\left[\text{quadriatic}\text{formula}\right]\\ x=\frac{12\pm \sqrt{96}}{6}\\ x=\frac{6\pm 2\sqrt{6}}{3}\end{array}$

Conclusion:

Hence the x intercepts is $\left(\frac{6+2\sqrt{6}}{3},0\right)$ and $\left(\frac{6-2\sqrt{6}}{3},0\right)$ .

(e)

To determine

To graph: f by using the results from part (a) to (d)

Answer to Problem 6CT

  

Explanation of Solution

Calculation:

Now plot the vertex, x-intercepts, y-intercept and axis of symmetry.

  

Conclusion:

Thus, the graph is drawn.