f is a trigonometric function of the form of a sin(bx+c) + d. Below is the graph f(x). The function intersects its midline at (3, -6.5) and has a maximum point at (4, - 2). Find a formula for f(x). Give an exact expression.​

See below for answer and explanation (and also a graph)

Step-by-step explanation:

For a sine function f(x) = a * sin(bx + c) + d:

Period: 2π/|b|

Phase shift: -c/b

Midline: d

Amplitude: |a|

The amplitude is the distance between the maximum and the midline, which is a = |-6.5-(-2)| = |-6.5+2| = |-4.5| = 4.5

The midline is already given to us as d=-6.5

The period would be the length of a cycle, which one finished cycle is defined at (4,-2) where the function must start at the maximum (0,-2) which is the distance of b, making the period of the function 2π/|4| = 2π/4 = π/2.

The last variable c can be found by plugging in a given point and solving for c. Plugging (4,-2) into the equation, you have -2=4.5sin(π/2(4)+c)-6.5 which gets you c=π/2+2πn for any integer n but we are only taking the first solution c=π/2

Therefore, your final equation is f(x) = 4.5sin(π/2x+π/2)-6.5

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step-by-step explanation:

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