Chapter 3.3, Problem 18AYU

In Problems 13-20, match each graph to one the following functions.

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

To determine

To calculate: The given graphs and matches the given equation $f(x)=x^2+2x+2$.

Answer to Problem 18AYU

The graph $B$ is the graph of the given equation.

Explanation of Solution

Given:

The given equation is $f(x)=x^2+2x+2$.

Formula Used:

The general form of a quadratic equation is

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

The graph of the above quadratic function is the graph of $f(x)=a{x}^2$ with $h$ units shifted horizontally and $k$ units shifted vertically.

The graph opens up if $a>0$ and it opens downward if $a<0$

If the given quadratic equation is $f(x)=a{x}^2+bx+c,a\ne 0$ then we can convert this into the standard form using $completingthesquare$ technique.

Thus, we have to divide the whole equation by $a$ and then add and subtract the square of half of the $x\text{-coefficient}$.

Therefore, we get

$\begin{array}{l}f(x)=x^2+\frac{b}{a}x+{\left(\frac{b}{2a}\right)}^2+\frac{c}{a}-{\left(\frac{b}{2a}\right)}^2\\ \Rightarrow f(x)={\left(x+\frac{b}{a}\right)}^2+\frac{c}{a}-{\left(\frac{b}{2a}\right)}^2\end{array}$

Calculation:

Here, we have to convert the given formula into the standard format.

Therefore, we have to use the transformation method to convert the given equation into the standard format.

Therefore, we get

$\begin{array}{l}f(x)=x^2+2x+2\\ \Rightarrow f(x)=x^2+2x+{\left(\frac{2}{2}\right)}^2+2-{\left(\frac{2}{2}\right)}^2\\ \Rightarrow f(x)=x^2+2x+1+2-1\\ \Rightarrow f(x)={\left(x+1\right)}^2+1\end{array}$

Here, in the given equation, we can see that $a=1,h=-1$ and $k=-1$

Here, since $a$ is positive, the graph opens upward.

The graph of the given function will be the graph of $f(x)=x^2$ with 1 unit shifted horizontally left and 1 unit shifted upward.

Therefore, the graph of the given function is