Question

Use a table of values to graph the given equation. State the domain and range.

$y=x^2+3x+1$

Short answer

The graph of the given function is

The domain is $\left(-\infty ,\infty \right)$and the range is $[-\mathbf{1}\mathbf{.25}\mathbf{,}\infty )$.

Step-by-step solution

Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$, where,$a\ne 0$ is called the standard form of the quadratic function.

Step 2. Define the domain and range of the function.

The domain is the set of all of the possible values of the independent variable $x$.

The range is the set of all the possible values of the dependent variable $y$.

Step 3. Calculate the table of values for the function y=x2+3x+1.

$x$	$y$
role="math" localid="1647698800192" $-2$	$-1$
role="math" localid="1647698810409" $-1$	$-1$
0	1
1	5
2	11

Step 4. Use the table of values to graph the function y=x2+3x+1.

Graph the ordered pairs from the table and connect them to create a smooth curve.

Step 5. State the domain and range for function y=x2+3x+1.

Observe the graph.

The parabola extends to infinity.

So, the domain is $\left(-\infty ,\infty \right)$.

The minimum value of the function is $-1.25$.

So, the range is $\left[-\right.1.25,\left.\infty \right)$.

Therefore, the domain is $\left(-\infty ,\infty \right)$and the range is $\left[-\right.1.25,\left.\infty \right)$.