Question

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.

Answer 1

The parabola has the following characteristics:

(i)
`Dom\{f(x)\} = \mathbb{R}`

(ii)
`Ran\{f(x)\} = (-\infty, 1]`

(iii) The parabola has an absolute maximum since
`C < 0`.

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

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:

`y-k = C\cdot (x-h)^(2)` (1)

Where:

`x` - Independent variable.

`y` - Dependent variable.

`h`,
`k` - Coordinates of the vertex.

`C` - Vertex parameter.

If we know that
`(x,y) = (2,-1)` and
`(h,k) = (3,1)`, then the vertex parameter of the parabola is:

`-1-1 = C\cdot (2-3)^(2)`

`-2 = C`

`C = -2`

According to this information, we find the following characteristics:

(i)
`Dom\{f(x)\} = \mathbb{R}`

(ii)
`Ran\{f(x)\} = (-\infty, 1]`

(iii) The parabola has an absolute maximum since
`C < 0`.

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