Question

Write the equation$y=x^2-4x+1$ in the form $y=a{\left(x-h\right)}^2+k$.

Short answer

The required form of the equation in$y=a{\left(x-h\right)}^2+k$ is $y={\left(x-2\right)}^2-3$ where$a=1,h=2\text{and}k=-3$.

Step-by-step solution

Step 1. Write down the given information.

The given equation is$y=x^2-4x+1$.

Step 2. Calculation.

The given equation $y=x^2-4x+1$ is converted to $y=a{\left(x-h\right)}^2+k$form as:

$\begin{array}{c}y=x^2-4x+1\\ =x^2-4x+1+{\left(-2\right)}^2-{\left(-2\right)}^2\mathrm{....}\left(\begin{array}{l}\text{Addandsubtract}\\ \text{squareof}{\left(-2\right)}^2\end{array}\right)\\ =\left(x^2-4x+{\left(-2\right)}^2\right)+\left(1-{\left(-2\right)}^2\right)\\ ={\left(x-2\right)}^2-3\end{array}$

Here $a=1,h=2\text{and}k=-3$.

Step 3. Conclusion.

The required form of the given equation is$y={\left(x-2\right)}^2-3$ where $a=1,h=2\text{and}k=-3$.