Chapter 3, Problem 16CR

To determine

To determine: The vertex form of the given equation. Also, find the vertex.

Answer to Problem 16CR

The required vertex form of the equation is $y={\left(x-1\right)}^2+19$ and the vertex is $\left(1,19\right)$ .

Explanation of Solution

Given information:

The equation: $y=x^2-2x+20$

Formula used:

The vertex form of a quadratic equation $a{x}^2+bx+c=0$ is $y=a{\left(x-h\right)}^2+k$ , where $\left(h,k\right)$ are the vertex of the equation.

Calculation:

The given equation is $y=x^2-2x+20$ .

To write the given equation in the vertex form $y=a{\left(x-h\right)}^2+k$ , use the method completing the square.

The middle term of the above equation can be written as:

  $2x=2\cdot x\cdot 1$

Here, $a=x$ and $b=1$

Find the value of $b^2$ .

  $\begin{array}{c}{b}^2=1^2\\ =1\end{array}$

Add 1 on each side of the above equation.

  $\begin{array}{c}y+1=\left(x^2-2x+1\right)+20\\ y+1=\left(x^2-2\cdot x\cdot 1+1^2\right)+20\\ y={\left(x-1\right)}^2+20-1\\ y={\left(x-1\right)}^2+19\end{array}$

This is the required vertex form of the given equation.

In the above equation, it is seen that $h=1$ and $k=19$ .

Thus, the vertex of the equation is $\left(1,19\right)$ .

Hence, the required vertex form of the equation is $y={\left(x-1\right)}^2+19$ and the vertex is $\left(1,19\right)$ .