To solve this problem, we need to follow the steps outlined in the question. I’ll guide you through each one. Since I can't use an actual calculator, I will describe the process step-by-step.

Step a) Plot the data

1. Input data: Enter the data into your calculator or software (such as a graphing calculator, or graphing software like Desmos or GeoGebra).

   • Let $x = 0$ correspond to the year 2010, and increment $x$ by 1 for each subsequent year.
   • Let $y$ represent the population of lynx in thousands.

2. Plot: Once the data points are entered, use your calculator's graphing tool to plot the points on a scatter plot.

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Step b) Find the sinusoidal function using regression

After plotting the data, use the regression tool in your calculator to find the sinusoidal function that best fits the data. The general form of a sinusoidal function is:

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

Where:

• $a$ is the amplitude (the distance from the middle of the wave to the peak),
• $b$ affects the period (how long it takes for the wave to complete one cycle),
• $c$ represents the phase shift (horizontal shift of the graph),
• $d$ is the vertical shift (the middle value of the wave).

Steps to use your calculator:

1. Use the regression feature of your calculator to select the sinusoidal model.
2. Once you input the data and choose sinusoidal regression, the calculator will give you the values of $a$, $b$, $c$, and $d$.
3. Record these values and write out the resulting sinusoidal equation in the form:

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

Make sure to round all values to the nearest tenth as specified.

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Step c) Predict the population in 2033

To predict the lynx population in the year 2033:

1. Since $x$ represents the number of years since 2010, for 2033, we calculate: $x = 2033 - 2010 = 23$

2. Substitute $x = 23$ into the sinusoidal model you obtained from part b). This will give you the population in thousands.

3. Multiply the result by 1,000 to convert the population into the actual number of lynx.

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Example

Let's assume the sinusoidal equation we obtained from regression is something like:

$y = 3.5 \sin\left(0.2(x - 5)\right) + 10$

For $x = 23$ (the year 2033):

$y = 3.5 \sin(0.2(23 - 5)) + 10$ $y = 3.5 \sin(0.2 \times 18) + 10$ $y = 3.5 \sin(3.6) + 10$

Using a calculator, $\sin(3.6) \approx -0.4425$:

$y = 3.5(-0.4425) + 10$ $y = -1.55 + 10 = 8.45$

So, the population in 2033 would be approximately 8.45 thousand lynx.

Finally, convert this to the actual number of lynx:

$8.45 \times 1,000 = 8,450$

Thus, the predicted population in 2033 is approximately 8,450 lynx.

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Let me know if you need more details or help with this!

Follow-up questions:

1. What are the key features of a sinusoidal function, and how do they relate to real-world data modeling?
2. How does the amplitude of a sinusoidal function affect the population model?
3. How do phase shifts affect the prediction of populations at different points in time?
4. Why might a sinusoidal function be a good fit for population data over time?
5. What are some other types of functions that could model periodic population changes, and how do they compare?

Tip:

Always double-check your regression equation to make sure it fits the data well before using it to make predictions!