A parabola has a vertex at (3, 1) and goes through the point (2, -1). Explain how you would write the equation of the parabola given this information. Then describe the domain and range. Additionally, discuss whether the function has a maximum or minimum, how you know this, and identify the value.

The parabola has the following characteristics:

(i)

(ii)

(iii) The parabola has an absolute maximum since .

(iv) The vertex parameter of the parabola is -2.

Step-by-step explanation:

Let suppose that parabola has a vertical axis of symmetry. From Analytical Geometry, the vertex form of the equation of the parabola is defined by the following formula:

(1)

Where:

- Independent variable.

- Dependent variable.

, - Coordinates of the vertex.

- Vertex parameter.

If we know that and , then the vertex parameter of the parabola is:

According to this information, we find the following characteristics:

(i)

(ii)

(iii) The parabola has an absolute maximum since .

(iv) The vertex parameter of the parabola is -2.