Q. 18 - Precalculus Enhanced with Graphing Utilities

Question

Match each graph to one the following functions.

$f\left(x\right)=x^2+2x+2$

Step-by-Step Solution

Answer

The required graph is in option D.

Step 1. Write given function in vertex form.

The given function is:

$f\left(x\right)=x^2+2x+2$

It can be written as:

$$\begin{gathered} f\left(x\right)=(x^2+2x+1)+1 \\ f\left(x\right)=(x+1{)}^2+1 [\because (a+b{)}^2=a^2+2ab+b^2] \end{gathered}$$

Step 2. Identify the graph of the function.

The vertex of the function $f\left(x\right)=a(x-h{)}^2+k$ is $(h,k)$. The graph opens up if $a>0$ and it opens down if $a<0$.

In the given function $a=1, h=-1, k=1$. So, the graph must be an upward parabola with vertex at $(-1,1)$.

Step 3. Conclusion.

The required graph is in option D.