Chapter 3, Problem 1RE

Solve each problem.

Write the function f(x) = 3x²– 2x + 1 in the form f(x) = a(x– h)² + k.

To determine

The function $f\left(x\right)=3x^2-2x+1$ in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$.

Answer to Problem 1RE

Solution:

The function $f\left(x\right)=3x^2-2x+1$ in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$ is $3{\left(x-\frac{1}{3}\right)}^2+\frac{2}{3}$.

Explanation of Solution

Consider the given function, $f\left(x\right)=3x^2-2x+1$.

Use the method of completing the square,

$\begin{array}{c}f\left(x\right)=3x^2-2x+1\\ =3\left(x^2-\frac{2}{3}x\right)+1\\ =3\left(x^2-\frac{2}{3}x+\frac{1}{9}-\frac{1}{9}\right)+1\\ =3\left(x^2-\frac{2}{3}x+\frac{1}{9}\right)-\frac{1}{3}+1\end{array}$

So, $f\left(x\right)=3{\left(x-\frac{1}{3}\right)}^2+\frac{2}{3}$

Therefore, the function $f\left(x\right)=3x^2-2x+1$ in the form $f\left(x\right)=a{\left(x-h\right)}^2+k$ is $3{\left(x-\frac{1}{3}\right)}^2+\frac{2}{3}$.