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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.

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Answer:

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.

User AabinGunz
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