Given the information about the minimum point $(-10.4, -9.8)$ and the maximum point $(-3.6, -1.2)$, we can infer several properties about the trigonometric function:

1. Amplitude

The amplitude $A$ of the function can be found by taking half the difference between the maximum and minimum values. The maximum value of the function is $-1.2$, and the minimum value is $-9.8$.

$A = \frac{|-1.2 - (-9.8)|}{2} = \frac{|-1.2 + 9.8|}{2} = \frac{8.6}{2} = 4.3$

2. Vertical Shift

The vertical shift $D$ is the average of the maximum and minimum values:

$D = \frac{-1.2 + (-9.8)}{2} = \frac{-11}{2} = -5.5$

So, the vertical shift is $-5.5$, which means the function oscillates around $y = -5.5$.

3. Period

To find the period $T$, we observe the horizontal distance between the minimum and maximum points. The distance between $-10.4$ and $-3.6$ is:

$T/2 = |-3.6 - (-10.4)| = |-3.6 + 10.4| = 6.8$

So, half of the period is 6.8, meaning the full period is:

$T = 2 \times 6.8 = 13.6$

4. Equation of the Function

The function could be represented as a sinusoidal function (either sine or cosine). Assuming the form of the cosine function $y = A \cos(Bx - C) + D$, where:

• $A = 4.3$ (amplitude),
• $B = \frac{2\pi}{T} = \frac{2\pi}{13.6}$,
• $D = -5.5$ (vertical shift).

Since we know the maximum occurs at $x = -3.6$, we can use this to find the phase shift $C$. The equation is then approximately:

$y = 4.3 \cos\left(\frac{2\pi}{13.6}(x + 3.6)\right) - 5.5$

Do you want to go deeper into any specific aspect of this, or ask additional questions?

Here are five relative questions you might consider:

1. How do we calculate the phase shift if the maximum point occurs at a different location?
2. How can this cosine equation be transformed into a sine equation?
3. What would be the effect of doubling the amplitude on the graph?
4. How can we verify that the calculated period matches the given points?
5. How does the vertical shift affect the real-world interpretation of the function?

Tip: Understanding the amplitude and vertical shift can help you recognize the range and midline of trigonometric functions when graphing.