Answer :
Answer:
The parabola has the following characteristics:
(i) [tex]Dom\{f(x)\} = \mathbb{R}[/tex]
(ii) [tex]Ran\{f(x)\} = (-\infty, 1][/tex]
(iii) The parabola has an absolute maximum since [tex]C < 0[/tex].
(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:
[tex]y-k = C\cdot (x-h)^{2}[/tex] (1)
Where:
[tex]x[/tex] - Independent variable.
[tex]y[/tex] - Dependent variable.
[tex]h[/tex], [tex]k[/tex] - Coordinates of the vertex.
[tex]C[/tex] - Vertex parameter.
If we know that [tex](x,y) = (2,-1)[/tex] and [tex](h,k) = (3,1)[/tex], then the vertex parameter of the parabola is:
[tex]-1-1 = C\cdot (2-3)^{2}[/tex]
[tex]-2 = C[/tex]
[tex]C = -2[/tex]
According to this information, we find the following characteristics:
(i) [tex]Dom\{f(x)\} = \mathbb{R}[/tex]
(ii) [tex]Ran\{f(x)\} = (-\infty, 1][/tex]
(iii) The parabola has an absolute maximum since [tex]C < 0[/tex].
(iv) The vertex parameter of the parabola is -2.