The graph shown represents a sinusoidal function, where the height $y$ of the sensor above the ground varies over time $t$ as the bicycle wheel spins. The key points of the graph can help derive the equation of the form $y(t) = A \sin(B(t - h)) + k$, where:

• $A$ is the amplitude.
• $B$ relates to the period.
• $h$ is the horizontal shift.
• $k$ is the vertical shift.

Step-by-step breakdown:

1. Amplitude ($A$):
   The amplitude is the distance from the midline to a maximum or minimum point. From the graph, the maximum value is 58 cm and the minimum value is 22 cm.

   • The midline (average of max and min):
     $\text{Midline} = \frac{58 + 22}{2} = 40 \, \text{cm}$
   • Amplitude:
     $A = 58 - 40 = 18 \, \text{cm}$

2. Period ($T$):
   The period is the time it takes to complete one full cycle. Based on the points given:

   • From $t = 0.75$ to $t = 3.75$, the graph completes half of a cycle (from a peak to the next peak), which corresponds to $3$ seconds. Thus, the full period $T$ is: $T = 6 \, \text{seconds}$
   • The value of $B$, which relates to the period, is given by: $B = \frac{2\pi}{T} = \frac{2\pi}{6} = \frac{\pi}{3}$

3. Vertical Shift ($k$):
   The midline of the function is at 40 cm, which represents the vertical shift $k$. Therefore, $k = 40$.

4. Horizontal Shift ($h$):
   The graph reaches its first peak at $t = 0.75$, so the horizontal shift is $h = 0.75$.

Equation:

Combining all the information, the equation becomes: $y(t) = 18 \sin\left( \frac{\pi}{3}(t - 0.75) \right) + 40$

This represents the sensor's height $y$ above the ground as a function of time $t$.

Would you like any details or further explanations?
Here are 5 related questions you might consider:

1. How would the equation change if the maximum height of the sensor increased?
2. What would the equation look like if the period of the spinning wheel changed to 8 seconds?
3. How do the sine and cosine functions differ when modeling this scenario?
4. Can this equation be written using a cosine function instead of sine?
5. How does the horizontal shift affect the timing of the peaks in the graph?

Tip: Always pay attention to the period when working with sinusoidal functions; it's key in understanding how the function repeats over time!