Question

A sinusoidal function whose period is π2 , maximum value is 10, and minimum value is −4 has a y-intercept of 10. what is the equation of the function described?

Answer 1

The equation of the sinusoidal function described with a period of π2, a maximum value of 10, and a minimum value of -4, having a y-intercept of 10, is y(x) = 7sin(4x) + 3.

Step-by-step explanation:

The student wants to know the equation of a sinusoidal function with a specific set of properties. Given the period (π2), maximum value (10), minimum value (-4), and y-intercept (10), we need to derive the equation. The amplitude (A) is half the distance between the maximum and minimum values, which is (10 - (-4))/2 = 7. The vertical shift (D) is the average of the maximum and minimum, so (10 + (-4))/2 = 3. The period (T) is given as π2, so the angular frequency (B) is 2π / T, which is 4. The y-intercept being the maximum value implies that the sine function starts at the peak, so the phase shift (C) is 0. The final equation of the wave function is y(x) = A sin(Bx + C) + D, which yields y(x) = 7sin(4x) + 3.

Answer 2

The equation of the sinusoidal function with the given parameters is y = 7 sin(4x) + 3.

Step-by-step explanation:

To find the equation of the sinusoidal function with a period of π/2, a maximum value of 10, a minimum value of -4, and a y-intercept of 10, we need to consider the parameters of the function: amplitude, period, phase shift, and vertical shift. The amplitude A is half the distance between the maximum and minimum values, so A = (10 - (-4))/2 = 7. The period P is given as π/2, which means the angular frequency ω = 2π/P = 2π/(π/2) = 4.

The vertical shift D is the average of the maximum and minimum, yielding D = (10 + (-4))/2 = 3. Since we know the function has a y-intercept of 10, and the general form is y = A sin(ωx + φ) + D, a y-intercept of 10 means that A + D = 10. Solving for the phase shift φ, we use φ = arcsin((10 - D) / A), which with our values gives φ = arcsin((10 - 3)/7). However, the sine function actually starts from zero, so there's no phase shift needed in this case (φ = 0).

Thus, the equation of the sinusoidal function is y = 7 sin(4x) + 3.