Q 52. - Precalculus Enhanced with Graphing Utilities

Question

Determine the quadratic function whose graph is given.

Step-by-Step Solution

Answer

The quadratic function is $f\left(x\right)=-2x^2-8x-2$.

Step 1. Given information.

The given graph is:

Step 2. Determine the vertex from the given graph.

By analyzing the graph of the function, the vertex of the function is $\left(-2,6\right)$.

So, $\left(h,k\right)=\left(-2,6\right)$.

The quadratic form of the function is:

$\begin{array}{rcl}f\left(x\right)& =& a{\left(x-h\right)}^2+k\\ f\left(x\right)& =& a{\left(x-\left(-2\right)\right)}^2+6\\ f\left(x\right)& =& a{\left(x+2\right)}^2+6\cdots \cdots \cdots \cdots \left(h=-2,k=6\right)\end{array}$

Step 3. Determine the value of a using the additional point 0 , - 2 .

Substitute $x=0$ and $f\left(0\right)=-2$ in the above function.

$\begin{array}{rcl}f\left(x\right)& =& a{\left(x+2\right)}^2+6\\ f\left(0\right)& =& a{\left(0+2\right)}^2+6\\ -2& =& 4a+6\\ 4a+6& =& -2\\ 4a& =& -2-6\\ 4a& =& -8\\ a& =& \frac{-8}{4}\\ a& =& -2\end{array}$

Step 4. Now, substitute a = - 2 in f x = a x + 2 2 + 6 .

$\begin{array}{rcl}f\left(x\right)& =& a{\left(x+2\right)}^2+6\\ & =& -2{\left(x+2\right)}^2+6\\ & =& -2\left(x^2+4x+4\right)+6\\ & =& -2x^2-8x-8+6\\ & =& -2x^2-8x-2\end{array}$

Step 5. Simplified answer.

Hence, the required quadratic function is $\begin{array}{rcl}f\left(x\right)& =& -2x^2-8x-2\end{array}$.