Respuesta :
Answer:
-2
Step-by-step explanation:
The midline of this function is y = [ -9 + 5 ] / 2 = -4/2 = -2
The amplitude of the function is abs value ( [-9 -5] /2) = abs value (-14/2 ) =abs value (-7) = 7
The period is twice as long as the regular csc function
The equation is
y = 7 csc (x/2) - 2
The equation of a cosecant function can be obtained by finding the variables in the general form of the function
[tex]The \ cosecant \ function \ equation \ is \ f(x) = \mathbf{7 \cdot csc\left(\dfrac{1}{2} \cdot x\right) - 2}[/tex]
The reason the above function is correct is as follows:
Question part missing: The part of the question that appear missing as obtained from a similar question is as follows;
The cosecant function range = (-∞, -9)U{5, ∞)
The given parameters are;
The consecutive asymptotes of the function are at x = 0, and x = 2·π
The function is not a reflection over the x-axis
The general form of the cosecant function is f(x) = A·csc(B·x + C) + D
Where:
A = The amplitude = (Maximum - Minimum)/2
The parent cosecant function range = (-∞, -1]U[1, ∞), a difference between maximum and minimum of 2, and having an amplitude of 2/2 = 1
The difference between the given maximum and minimum is 5 - (-9) = 14
∴ The amplitude, A = (5 - (-9))/2 = 14/2 = 7
A = 7
B: The period factor
The consecutive asymptote in the parent function, represent half a period
Therefore;
The period, T = 2·π/B = 2 × 2·π = 4·π
B = 2·π/4·π = 1/2
B = 1/2
C: Horizontal shift
Given that x = 0 is an asymptote, we have;
csc(B×0 + C) = ∞
csc(C) = ∞
1/csc(C) = 1/∞ = 0
sin(C) = 0
C = 0
D: Vertical shift
The vertical shift, D = (Maximum + Minimum)/2
D = (-9 + 5)/2 = -2
D = -2
Therefore, the equation of the cosecant function is given by plugging in the values of A, B, C, and D as follows;
f(x) = 7·csc((1/2)·x + 0) + (-2)
[tex]f(x) = \mathbf{7 \cdot csc\left(\dfrac{1}{2} \cdot x\right) - 2}[/tex]
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