Answer:
[tex]f(x)=6\sin\left(\dfrac{\pi}{4}x\right)-10[/tex]
Step-by-step explanation:
The general form of a sinusoidal function is:
[tex]f(x)=\text{A} \sin\left(\text{B}(x+\text{C})\right)+\text{D}[/tex]
where:
- A = amplitude (height from the midline to the peak or trough)
- 2π/B = period (horizontal distance between consecutive peaks)
- C = phase shift (horizontal shift where positive is to the left)
- D = vertical shift (y = D is the midline)
[tex]\dotfill[/tex]
Midline
The midline of a sinusoidal function is a horizontal line that lies equidistant between the function's maximum and minimum values.
If the graph of a sinusoidal function intersects its midline at (0, -10), then the midline is y = -10. Therefore:
[tex]\rm D = -10[/tex]
[tex]\dotfill[/tex]
Amplitude
The amplitude of a sinusoidal function is the distance from the midline to the maximum or minimum value of the function.
Given that the function has a maximum value of y = -4, the amplitude (A) of the function is:
[tex]\rm A=4 - (-10) \\\\A= 6[/tex]
[tex]\dotfill[/tex]
Period
The horizontal distance between the point at which the function intersects its midline (0, -10) and the maximum point at (2, -4) is one-quarter its period. Therefore, the period is:
[tex]\rm Period = 4 \times (2-0)\\\\Period=8[/tex]
Substitute the period into the period formula and solve for B:
[tex]\rm \dfrac{2\pi}{B}=8\\\\\\B=\dfrac{2\pi}{8}\\\\\\B=\dfrac{\pi}{4}[/tex]
[tex]\dotfill[/tex]
Phase Shift
The parent sine function y = sin(x) crosses its midline (x-axis) at x = 0 and then curves upward to its maximum. Since this function exhibits the same behavior, it has not undergone a horizontal shift, so:
[tex]\rm C = 0[/tex]
[tex]\dotfill[/tex]
Formula of the function
Substituting A = 6, B = π/4, C = 0 and D = -10 into the general form gives:
[tex]f(x)=6\sin\left(\dfrac{\pi}{4}(x+0)\right)-10\\\\\\f(x)=6\sin\left(\dfrac{\pi}{4}x\right)-10[/tex]
Therefore, the formula of the function where x is entered in radians is:
[tex]\Large\boxed{\boxed{f(x)=6\sin\left(\dfrac{\pi}{4}x\right)-10}}[/tex]
