To solve for the function $y = \cos(x)$ using the points $(275, 175)$, $(400, 275)$, and $(500, 175)$, we'll follow a systematic approach to determine the transformation of the cosine function that fits these points.

Step 1: Understand the Cosine Function

The general form of the cosine function is: $y = a \cos(bx + c) + d$ where:

• $a$ is the amplitude.
• $b$ affects the period.
• $c$ is the phase shift.
• $d$ is the vertical shift.

Step 2: Analyze the Given Points

• Point 1: $(275, 175)$
• Point 2: $(400, 275)$
• Point 3: $(500, 175)$

Step 3: Identify Key Features from the Points

• The $y$ value at $(400, 275)$ is the maximum, which suggests that this point represents the peak of the cosine wave.
• The $y$ values at $(275, 175)$ and $(500, 175)$ are identical, suggesting that these points might correspond to positions where the cosine function has the same value, indicating symmetry.

Step 4: Determine the Amplitude and Vertical Shift

1. Amplitude $a$:

   • The amplitude is half the difference between the maximum and minimum $y$ values. $a = \frac{275 - 175}{2} = 50$

2. Vertical Shift $d$:

   • The vertical shift is the average of the maximum and minimum $y$ values. $d = \frac{275 + 175}{2} = 225$

So far, our function is: $y = 50 \cos(bx + c) + 225$

Step 5: Determine the Period and Phase Shift

• The points $(275, 175)$ and $(500, 175)$ correspond to $y$ values equal to the mean, suggesting these points might be a quarter and three-quarters through the period of the cosine function.
• The full period $T$ occurs between the first and third points, covering a $2\pi$ range.

Given:

• $T = 500 - 275 = 225$

The period formula is: $\text{Period} = \frac{2\pi}{b}$

So: $b = \frac{2\pi}{225}$

Step 6: Determine the Phase Shift $c$

• The maximum point occurs at $x = 400$. For a standard cosine function, this corresponds to $x = 0$, so the phase shift should make $bx + c = 0$ at $x = 400$.

So: $400 \cdot \frac{2\pi}{225} + c = 0$ $c = - \frac{800\pi}{225}$

Final Equation

Putting everything together, the cosine function that fits the given points is approximately: $y = 50 \cos\left(\frac{2\pi}{225} x - \frac{800\pi}{225}\right) + 225$

Summary:

The final function is: $y = 50 \cos\left(\frac{2\pi}{225} (x - 400)\right) + 225$ This function describes the cosine wave that fits the given points.

Do you have any questions or would you like to see more details?

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Related Questions:

1. How do you derive the amplitude of a trigonometric function from points?
2. What steps are involved in determining the period of a cosine function from given points?
3. How do you find the vertical shift in a trigonometric function?
4. How does phase shift affect the graph of a cosine function?
5. How can the cosine function be used to model real-world periodic phenomena?
6. What is the significance of the period $T$ in a trigonometric function?
7. How does changing the amplitude affect the appearance of the cosine graph?
8. What role does the vertical shift $d$ play in adjusting the baseline of the graph?

Tip: When adjusting trigonometric functions to fit data points, carefully observe the symmetry and key points, like maximums and minimums, to deduce transformations.