Chapter 1.7, Problem 47E

Solution

To write: The function $y=2x^2+24x+25$ in vertex form and identify the vertex.

The vertex form of the function is $y=2{\left(x+6\right)}^2-47$ and the vertex is $\left(-6,-47\right)$ .

Given information:

The function, $y=2x^2+24x+25$ .

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:

  $\begin{array}{c}y=2x^2+24x+25\\ =2\left(x^2+12x\right)+25\\ =2\left(x^2+2\cdot x\cdot 6+{\left(6\right)}^2-{\left(6\right)}^2\right)+25\\ =2{\left(x+6\right)}^2-2{\left(6\right)}^2+25\\ =2{\left(x+6\right)}^2-2\cdot 36+25\\ =2{\left(x+6\right)}^2-72+25\\ =2{\left(x+6\right)}^2-47\end{array}$

Now, compare the function $y=2{\left(x+6\right)}^2-47$ with $y=a{\left(x-h\right)}^2+k$ to get the vertex as $\left(-6,-47\right)$ .

Conclusion:

The vertex form of the function is $y=2{\left(x+6\right)}^2-47$ and the vertex is $\left(-6,-47\right)$ .