Chapter 8.3, Problem 11E

(a)

To determine

To find: The axis of symmetry of the graph of the function.

Answer to Problem 11E

The axis of symmetry is $x=5$ .

Explanation of Solution

Given information: $f(x)=\frac{2}{5}{x}^2-4x+14$

Calculation:The axis of symmetry of the quadratic function $f(x)=a{x}^2+bx+c$ is $x=-\frac{b}{2a}$ .

The axis of symmetry when $a=\frac{2}{5}$ and $b=-4$ is given by,

  $\begin{array}{l}x=-\frac{b}{2a}\\ \Rightarrow x=-\frac{(-4)}{2\cdot (\frac{2}{5})}\\ \Rightarrow x=\frac{20}{4}\\ \Rightarrow x=5\end{array}$

The axis of symmetry is $x=5$ .

(b)

To determine

To find:The vertex of the graph of the function.

Answer to Problem 11E

The vertex is $(5,4)$ .

Explanation of Solution

Given information: $f(x)=\frac{2}{5}{x}^2-4x+14$

Calculation:The axis of symmetry is $x=5$ , so the x -coordinate of the vertex is $5$ .

Using the function to find the y -coordinate of the vertex,

  $\begin{array}{l}f(x)=\frac{2}{5}{x}^2-4x+14\\ \Rightarrow f(5)=\frac{2}{5}\cdot 5^2-4\cdot 5+14\\ \Rightarrow f(5)=10-20+14\\ \Rightarrow f(5)=4\end{array}$

The vertex is $(5,4)$ .