Question

Designer functions Design a sine function with the given properties. It has a period of 12 hr with a minimum value of -4 at $t=0$ hr and a maximum value of 4 at $t=6\mathrm{hr}.$

Short answer

Question: Give the sine function that represents a wave with a period of 12 hours, a minimum value of -4 at $t=0$, and a maximum value of 4 at $t=6$. Answer: The sine function that represents the wave is: $$f(t)=4\sin (\frac{\pi }{6}(t+3))$$

Step-by-step solution

Identify the general sine function with variables

The general sine function with variables is given by: $$f(t)=A\sin B(t-C)+D$$ Here, A = amplitude B = frequency, which is related to the period by $B=\frac{2\pi }{\text{Period}}$ C = phase shift D = vertical displacement We need to find the values of A, B, C, and D that satisfy the given conditions.

Determine the amplitude

The sine function has a minimum value of -4 and a maximum value of 4. The amplitude can be calculated as half the difference between the maximum and minimum values. Therefore, Amplitude (A) = $\frac{4-(-4)}{2}=\frac{8}{2}=4$.

Determine the period and frequency

The given period of the sine function is 12 hours. Using the formula $B=\frac{2\pi }{\text{Period}}$, we can find the frequency (B). Frequency (B) = $\frac{2\pi }{12}=\frac{\pi }{6}$.

Determine the phase shift

The minimum value of the sine function occurs at $t=0$. For the standard sine function, the minimum value occurs at 3 hours, i.e., $t=\frac{3\pi }{2}$. Thus, we need to shift the minimum value by $\frac{3\pi }{2}$ to the left. The phase shift can be found by equating the argument of the sine function to $\frac{3\pi }{2}$: $B(t-C)=\frac{3\pi }{2}$ Plugging in the value of B from step 3: $\frac{\pi }{6}(t-C)=\frac{3\pi }{2}$ Now, we know that the minimum value occurs at $t=0$. Therefore, substituting $t=0$: $\frac{\pi }{6}(-C)=\frac{3\pi }{2}$ Solving for C: Phase Shift (C) = $-3\text{ hours}$.

Determine the vertical displacement

The vertical displacement (D) can be determined from the sum of the maximum value and the minimum value of the function, divided by 2: Vertical Displacement (D) = $\frac{4+(-4)}{2}=0$.

Combine all values to find the designed sine function

Substituting the values of amplitude (A), frequency (B), phase shift (C), and vertical displacement (D) in the general sine function, we get: $$f(t)=4\sin (\frac{\pi }{6}(t+3))+0$$ So, the designed sine function with the given properties is: $$f(t)=4\sin (\frac{\pi }{6}(t+3))$$

Key concepts

Amplitude of Sine Function

The amplitude of a sine function measures the height of each wave from the central axis (where the function crosses zero) to its peak. In other words, it determines how 'tall' or 'short' the waves of the sine function are. For instance, if the maximum and minimum values of the sine function are 4 and -4 respectively, the amplitude can be found by taking the difference between these two values and dividing it by 2.

Mathematically, this can be expressed as:
Amplitude (A) = $\frac{\text{{maximum value}}-\text{{minimum value}}}{2}$ = $\frac{4-(-4)}{2}=\frac{8}{2}=4$. This value tells us that the sine wave will rise 4 units above and drop 4 units below the horizontal axis.

Frequency and Period Relationship

The frequency and period of a sine function have an inverse relationship. The frequency (B) represents how many cycles the function completes in a unit of time, while the period is the duration of time it takes to complete one full cycle. You can calculate the frequency by taking the reciprocal of the period and multiplying it by $2\pi$.

This can be seen in the formula:
$B=\frac{2\pi }{\text{Period}}$. For example, if the period is 12 hours, the frequency is $\frac{2\pi }{12}=\frac{\pi }{6}$. Higher frequency means the waves are closer together, whereas a larger period means the waves are more spread out.

Phase Shift Calculation

Phase shift in a sine function indicates a horizontal shift from the standard position of the wave. Positive phase shift means the wave is shifted to the right, while negative means it has been shifted to the left. The 'C' in the function $f(t)=A\sin B(t-C)+D$ represents the phase shift.

To calculate the phase shift, you can set the argument inside the sine function during the point of minimum or maximum value and solve for C. Following the provided example, if the minimum value occurs at 0 hours instead of at the expected $\frac{3\pi }{2}$ for a standard sine wave, we calculate the shift by:
$B(t-C)=\frac{3\pi }{2}$.
Substituting B and solving for C when t is 0 hours gives the phase shift of -3 hours, indicating a leftward shift of the whole function by 3 hours.