Chapter 7.1, Problem 3QR

To determine

To state: All the values of x (if any) at which the function $x^2-2x+3\text{on}\left[-4,2\right]$ changes sign on the given interval then sketch a number line graph if the interval, and indicate the sign of the function on each subinterval.

Answer to Problem 3QR

The function does not change sign in the given interval and the line graph is:

  

Explanation of Solution

Given information:

The given function is $x^2-2x+3\text{on}\left[-4,2\right]$ .

Now consider the given function,

  $x^2-3x+2\text{on}\left[-2,4\right]$

First, find he solution of the equation $x^2-2x+3=0$ , then use the obtained values to get the partitioning interval.

  $x^2-2x+3=0$

Now take $b^2-4ac$ and substitute $b=-2,a=1,\text{and}c=3$ ,

  $\begin{array}{c}{\left(-2\right)}^2-4\left(1\right)\left(3\right)=4-12\\ =-8\\ -8<0\end{array}$

Therefore, $x^2-2x+3=0$ has no solution and ${\left(x-1\right)}^2+2>0$ for all x in the given interval.

Therefore, the function does not change sign in the given interval.

The line graph is: