Question

The function g is a trigonometric function of the form (g(x)=acos(bx+c)+d). Below is the graph of g(x). The function has a maximum point at ((3pi/4), -2) and a minimum point at ((2pi), -14). Find a formula for g(x), giving an exact expression.

a) (g(x) = -12cosleft(pi/4x - pi/2) + 6
b) (g(x) = 6cosleft(pi/4x + pi/2) - 12
c) (g(x) = -2cosleft(pi/4x - pi/2) - 14
d) (g(x) = -14cosleft(pi/4x + pi/2) + 2

Answer 1

The correct formula for the function g(x) is g(x) = -2cos(π/4x - π/2) - 14.

Step-by-step explanation:

Given that the function g(x) has a maximum point at (3π/4, -2) and a minimum point at (2π, -14), we can determine the values of a, b, c, and d in the general form g(x) = acos(bx + c) + d. The amplitude of the cosine function, a, is the vertical distance from the maximum or minimum point to the centerline. In this case, a = (14 - (-2))/2 = 8.

The period of the cosine function, T, is the horizontal distance for one complete cycle, and T = 2π/(π/4) = 8. Therefore, b = 2π/8 = π/4. The phase shift, c, is determined by the horizontal shift of the cosine function, and in this case, c = -π/2. Finally, d is the vertical shift, and d = -14. Plugging these values into the general form gives the correct formula g(x) = -2cos(π/4x - π/2) - 14.

Option (c) is the accurate choice, as it matches the determined formula. Option (a) has the wrong coefficient for the cosine function, and options (b) and (d) have incorrect signs and coefficients. Therefore, the correct answer is (c).