Question

Use a table of values to graph the following functions. State the domain and range.

$\mathit{y}\mathbf{=}{\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{x}\mathbf{+}\mathbf{5}$

Short answer

The graph of the function is:

The domain is all real numbers and the range is $\left\{y|y\ge 4\right\}$.

Step-by-step solution

Step 1. Define the concept.

The domain of a function is the complete set of possible values of the independent variable. The domain is the set of all possible $x$-values that will make the function "work" and will output real $y$-values.

Therangeof a function is the complete set of all possibleresulting valuesof the dependent variable ($y$,usually) after we have substituted the domain.

Step 2. Make a table to calculate points on the graph. 

For the equation,$y=x^2+2x+5$,make a table to get some points on the graph.

$x$	$y=x^2+2x+5$	$(x,y)$
0	$y=0^2+2\left(0\right)+5=5$	$(0,5)$
role="math" localid="1648204417494" $-2$	$y={(-2)}^2+2(-2)+5=5$	$(-2,5)$
$-5$	$y={(-5)}^2+2(-5)+5=20$	$(-5,20)$
1	$y={\left(1\right)}^2+2\left(1\right)+5=8$	$(1,8)$
3	$y={\left(3\right)}^2+2\left(3\right)+5=20$	$(3,20)$

Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity.

Step 3. Graph the equation and write domain and range.

Graph the ordered pairs, and connect them to create a smooth curve. The parabola extends to infinity.

The domain is all real numbers. The range is all real numbers greater than or equal to the minimum value or $\left\{y|y\ge 4\right\}$.