Q. 50 - 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 $f\left(x\right)=-(x-2{)}^2+3$ or $f\left(x\right)=-x^2+4x-1$.

Step 1. Given information.

The given graph is:

Step 2. Determine the equation.

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

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

To determine the value of $a$, we use the fact that $f\left(0\right)=-1$ (the y-intercept).

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

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

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

Step 3. Conclusion.

The quadratic function whose graph is given is $f\left(x\right)=-(x-2{)}^2+3$ or $f\left(x\right)=-x^2+4x-1$.