Chapter 8, Problem 10CR

To determine

Tograph:The function $y\left(x\right)=x^2-2x+7$ and describe its domain and range.

Explanation of Solution

Given information:

The given function to graph is $y\left(x\right)=x^2-2x+7$ , of which domain and range is to be determined.

Formulaused:

The standard form of the quadratic function is $f\left(x\right)=a{\left(x-h\right)}^2+k$ whose vertex is $\left(h,k\right)$ , and the domain is set of all real numbers or $\left(-\infty ,\infty \right)$ and range $y\ge k$ .

Graphs:

The standard form of given function is $y\left(x\right)=\left(x^2-2x+1\right)+6$ or $y\left(x\right)={\left(x-1\right)}^2+6$ .

The graphs of function $y\left(x\right)=x^2-2x+7$ is shown in Figure-1 here.

Calculation: From the equation, $y\left(x\right)={\left(x-1\right)}^2+6$ and the graph, it is clear that for each real there exists a y. Hence, the domain of function is the set of all real numbers or $\left(-\infty ,\infty \right)$ .

Since, the minima of yis 6, i.e., $y\ge 6$ . Thus, the range of the function $y\left(x\right)=x^2-2x+7$ is $y\ge 6$ .

Moreover, $y\left(x\right)={\left(x-1\right)}^2+6$

  $\Rightarrow x=1\pm \sqrt{y-6}$ , thus there exists a real $x\in \left(-\infty ,\infty \right)$ for $y\ge 6$ .