Q. 6 - Precalculus Enhanced with Graphing Utilities

Question

Determine the quadratic function for the given graph. 

Step-by-Step Solution

Answer

The quadratic function is $f\left(x\right)=2{(x-1)}^2-32$.

Step 1. Given information.

Determine the quadratic function for the given graph.

Step 2. Find the quadratic function.

If the vertex $\left(h,k\right)$ and one point on the graph of a quadratic function are known, then we can write

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

From the given graph, vertex is $\left(1,-32\right) \mathrm{so}, \mathrm{h}=1 \mathrm{and} \mathrm{k}=-32$

Now substitute these values into the equation

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

To determine the value of  $a$, use a fact that point$\left(5,0\right)$ is on the graph. 

Step 3. substitute the values.

$$\begin{gathered} f\left(5\right)=a{\left(5-1\right)}^2-32 \\ 0=16a-32 \\ 16a=32 \\ a=2 \end{gathered}$$

Substitute the value of $a$ into the function.

$f\left(x\right)=a{(x-h)}^2+k=2{(x-1)}^2-32$

Step 4. Conclusion.

The quadratic function is $f\left(x\right)=2{(x-1)}^2-32$.