Chapter 1.7, Problem 13Q

Solution

To write: The function $y=x^2-4x+9$ in vertex form and identify the vertex.

The vertex form of the function is $y={\left(x-2\right)}^2+5$ and the vertex is $\left(2,5\right)$ .

Given information:

The function, $y=x^2-4x+9$ .

Concept used:

Vertex form a quadratic function is given as $y=a{\left(x-h\right)}^2+k$ , where $\left(h,k\right)$ is the vertex of the quadratic function.

Calculation:

Write the function in the vertex form as follows:

  $\begin{array}{c}y=x^2-4x+9\\ y=x^2-2\cdot x\cdot 2+2^2-2^2+9\\ y={\left(x-2\right)}^2-4+9\\ y={\left(x-2\right)}^2+5\end{array}$

Now, compare the function $y={\left(x-2\right)}^2+5$ with $y=a{\left(x-h\right)}^2+k$ to get the vertex as $\left(2,5\right)$ .

Conclusion:

The vertex form of the function is $y={\left(x-2\right)}^2+5$ and the vertex is $\left(2,5\right)$ .