Q. 49 - Precalculus Enhanced with Graphing Utilities

Question

Determine the quadratic function whose graph is given.

Step-by-Step Solution

Answer

The quadratic function whose graph is given is $f\left(x\right)=-(x+3{)}^2+5$ or $f\left(x\right)=-x^2-6x-4$.

Step 1. Given information.

The given graph is:

Step 2. Determine the equation.

The vertex is $(-3,5)$, so $h=-3$ and $k=5$. Substitute these values into equation $f\left(x\right)=a(x-h{)}^2+k$.

$f\left(x\right)=a(x+3{)}^2+5$

To determine the value of $a$, we use the fact that $f\left(0\right)=-4$ (the $y$-intercept).

$\begin{array}{rcl}-4& =& a(0+3{)}^2+5\\ -4-5& =& 9a\\ -9& =& 9a\\ -1& =& a\end{array}$

Substitute $a=-1$ in $f\left(x\right)=a(x+3{)}^2+5$.

$$\begin{gathered} f\left(x\right)=-1(x+3{)}^2+5 \\ f(x)=-(x+3{)}^2+5 \\ f\left(x\right)=-x^2-6x-9+5 \\ f\left(x\right)=-x^2-6x-4 \end{gathered}$$

Step 3. Conclusion.

The quadratic function whose graph is given is $f\left(x\right)=-(x+3{)}^2+5$ or $f\left(x\right)=-x^2-6x-4$.