Q. 20 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 20 and 21, find the quadratic function for which: 

Vertex is $\left(-1,2\right)$; contains the point $\left(1,6\right)$.

Step-by-Step Solution

Answer

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

Step 1. Given information.

Find the quadratic function for which: 

Vertex is $\left(-1,2\right)$; contains the point $\left(1,6\right)$.

Step 2. Find quadratic function.

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

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

The given vertex is $\left(-1,2\right) \mathrm{so}, h=-1, k=2$

now substitute these values into the equation:

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

To determine the value of $a$ , use a fact that point $\left(1,6\right)$ is on the graph.

So, $f\left(1\right)=6$

Step 3. substitute the values.

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

Substitute the value of $a$ into the function.

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

Step 4. Conclusion.

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