Q. 48 - 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 $f\left(x\right)=(x-2{)}^2+1$ or $f\left(x\right)=x^2-4x+5$.

Step 1. Given information.

The given graph is:

Step 2. Determine the equation.

The vertex is $(2,1)$, so $h=2$ and $k=1$. Substitute these values into equation $f\left(x\right)=a(x-h{)}^2+k$.

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

To determine the value of $a$, we use the fact that $f\left(0\right)=5$ (the y-intercept).

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

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

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

Step 3. Conclusion.

The quadratic function whose graph is given is $f\left(x\right)=(x-2{)}^2+1$ or $f\left(x\right)=x^2-4x+5$.