To find the equation of a sinusoidal function, we need to determine values for $a$, $b$, $c$, and $d$ in the general form:

$y = a \sin(b(x - c)) + d$

or

$y = a \cos(b(x - c)) + d$

Step-by-Step Solution

1. Determine $d$ (the vertical shift):

   • The central axis is at $y = 5$, so $d = 5$.

   $d = 5$

2. Determine $a$ (the amplitude):

   • The maximum distance from the central axis to a minimum (or maximum) point is the amplitude.
   • Given that the minimum value is $y = -1$ and the central axis is $y = 5$, the amplitude is the distance from 5 to -1.

   $a = |5 - (-1)| = 6$

3. Determine $b$ (the frequency):

   • The period $T$ of the sinusoidal function can be found from two consecutive minimums at $x = 3$ and $x = 13$.
   • The distance between these points is $T = 13 - 3 = 10$.
   • Since the period $T$ is related to $b$ by the formula $T = \frac{2\pi}{b}$, we can solve for $b$:

   $b = \frac{2\pi}{T} = \frac{2\pi}{10} = \frac{\pi}{5}$

4. Determine $c$ (the phase shift):

   • Since the minimum occurs at $x = 3$, we can set this as the point where $x = 3$ aligns with the minimum of a cosine function, as cosine starts at a maximum or minimum.
   • For a cosine function with a minimum at $x = 3$, $c = 3$.

   $c = 3$

5. Write the equation:

   • Using these values, we get:

$y = -6 \cos\left(\frac{\pi}{5}(x - 3)\right) + 5$

Final Answer:

The equation of the sinusoidal graph is:

$y = -6 \cos\left(\frac{\pi}{5}(x - 3)\right) + 5$

This equation meets all the conditions specified for the sinusoidal function.

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Would you like further details or have any questions?

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Here are five related questions to consider:

1. How would the equation change if the maximum point instead of the minimum was given?
2. What is the effect of changing the amplitude on the shape of the graph?
3. How does altering the phase shift $c$ affect the graph's position?
4. What happens to the graph if the central axis $d$ is modified?
5. How can we determine the equation if given maximum points instead of minimum points?

Tip: When analyzing sinusoidal functions, remember that sine and cosine functions can model similar behaviors but may require adjustments in phase shift.