Chapter 8.5, Problem 94E

(a)

To determine

To find: The intercept form of the quadratic function $g(x)=2{(x-1)}^2-2$.

Answer to Problem 94E

  $g(x)=2(x-0)(x-2)$

Explanation of Solution

Given information: $g(x)=2{(x-1)}^2-2$

Calculation:The intercept form of quadratic function is given by the form,

  $g(x)=a(x-p)(x-q)$ where $a\ne 0$ .

  $\begin{array}{l}$ g(x)=2{(x-1)}^2-2$\Rightarrow g(x)=2(x^2-2x+1)-2\\ \Rightarrow g(x)=2x^2-4x+2-2\\ \Rightarrow g(x)=2x^2-4x\\ \Rightarrow g(x)=2x(x-2)\\ \Rightarrow g(x)=2(x-0)(x-2)\end{array}$

Thus, the intercept form is $g(x)=2(x-0)(x-2)$ .

(b)

To determine

To graph: The function $g(x)=2{(x-1)}^2-2$.

Explanation of Solution

Given information: $g(x)=2{(x-1)}^2-2$

Graph:

  

The function $g(x)=2{(x-1)}^2-2$ in intercept form is,

  $g(x)=2(x-0)(x-2)$ .

The x -intercepts are $p=0$ and $q=2$ , so we plot the points $(0,0)$ and $(2,0)$ .

The axis of symmetry is $x=\frac{p+q}{2}=\frac{0+2}{2}=\frac{2}{2}=1$ .

Now, we plot the axis of symmetry $x=1$ .

Then, we find the vertex.

The x -coordinate of the vertex is 1. To find the y -coordinate of the vertex, we substitute $x=1$ .

  $\begin{array}{l}g(x)=2(x-0)(x-2)\\ \Rightarrow g(1)=2(1-0)(1-2)\\ \Rightarrow g(1)=2\cdot 1\cdot -1\\ \Rightarrow g(1)=-2\end{array}$

So, the vertex is $(1,-2)$ .

Now, we draw a parabola through the vertex $(1,-2)$ and the points where x -intercepts occur, that is, $(0,0)$ and $(2,0)$ .