Chapter 5, Problem 1GR

Solution

To graph: the quadratic function.

Given:

Function:

  $f(x)=x^2-8x+7$

Concept use:

  $\left.1.\right)$ To graph quadratic function convert it into a standard form using completing the square method.

  $\left.2.\right)$ The standard form of parabola is

  $y=a{\left(x-h\right)}^2+k$

Where $\left(h,k\right)$ is vertex of parabola.

Also

If $a>0$ then parabola opens upward, and vertex is minimum.

If $a<0$ then parabola opens downward and vertex is maximum.

Calculation:

  $\begin{array}{l}f(x)=x^2-8x+7\\ f\left(x\right)=x^2-\left(2\times 4\right)x+7\end{array}$

Add and subtracts $9$ then

  $\begin{array}{l}f\left(x\right)=x^2-\left(2\times 4\right)x+7+9-9\\ f\left(x\right)=x^2-\left(2\times 4\right)x+16-9\\ f\left(x\right)={\left(x-4\right)}^2-9\end{array}$

Compare with standard form then

Vertex $=\left(4,-9\right)$

Also $a=1$ which is greater than zero.

Thus, parabola is opens upward with minimum is vertex.

Now graph of function is