# Sinusoidal Modeling Common Core Algebra Ii Homework Answers

## Sinusoidal Modeling: A Complete Guide for Common Core Algebra II Homework

Sinusoidal modeling is a technique that uses sine and cosine functions to describe periodic phenomena, such as sound waves, tides, temperature cycles, and more. Sinusoidal models can help us understand the patterns and behaviors of these phenomena, as well as make predictions and solve problems.

## sinusoidal modeling common core algebra ii homework answers

In this article, we will explain what sinusoidal models are, how to graph them, how to find their parameters, and how to use them in common core algebra II homework. We will also provide some examples and exercises for you to practice.

## What are sinusoidal models?

A sinusoidal model is a function of the form:

y = a \cdot \cos(b \cdot x + c) + d or y = a \cdot \sin(b \cdot x + c) + d

where a, b, c, and d are constants that determine the shape and position of the graph. The graph of a sinusoidal model is a smooth curve that repeats itself at regular intervals. This interval is called the period of the function.

The constants a, b, c, and d have specific meanings in the context of sinusoidal modeling. Here is a summary of what they represent:

a is the amplitude of the function. It measures how far the graph goes above and below the horizontal line y = d. The amplitude is always positive.

b is the angular frequency of the function. It measures how fast the graph oscillates. The angular frequency is related to the period by the formula b = \frac2\piT, where T is the period.

c is the phase shift of the function. It measures how much the graph is shifted horizontally from the standard position. The phase shift can be positive or negative.

d is the vertical shift of the function. It measures how much the graph is shifted vertically from the x-axis. The vertical shift can be positive or negative.

## How to graph sinusoidal models?

To graph a sinusoidal model, we need to identify the values of a, b, c, and d from the given equation. Then, we can use these values to find some key points on the graph, such as the maximum, minimum, midline, and x-intercepts. Finally, we can sketch a smooth curve that passes through these points and repeats itself periodically.

Here are some steps to follow:

Determine the amplitude a. This is half of the vertical distance between the maximum and minimum points on the graph.

Determine the period T. This is the horizontal distance between two consecutive peaks or troughs on the graph. Use the formula T = \frac2\pib to find it.

Determine the phase shift c. This is how much the graph is shifted left or right from its standard position. Use the formula c = -\frac\textphase shiftb to find it.

Determine the vertical shift d. This is how much the graph is shifted up or down from the x-axis. This is also the y-coordinate of the midline of the graph.

Find some key points on the graph using these values. For example, you can find:

The maximum point: This occurs when x = -\fraccb. The y-coordinate is y = a + d.

The minimum point: This occurs when x = -\fraccb + \fracT2. The y-coordinate is y = -a + d.

The midline: This is a horizontal line with equation y = d. The graph crosses this line at intervals of \fracT4.

The x-intercepts: These are points where y = 0. Depending on whether you have a cosine or sine function, you can use different formulas to find them.

If you have a cosine function, then one x-intercept occurs when x = -\fraccb + \fracT4. The other x-intercepts occur at intervals of \fracT2.

If you have a sine function, then one x-intercept occurs when

## How to use sinusoidal models in common core algebra II homework?

Sinusoidal models can be used to answer various questions and problems related to periodic phenomena. For example, you can use sinusoidal models to:

Find the value of a variable at a given time or position.

Find the time or position when a variable reaches a certain value.

Find the maximum or minimum value of a variable and when it occurs.

Find the average value of a variable over a given interval.

Compare and contrast different sinusoidal models and their parameters.

To use sinusoidal models in common core algebra II homework, you need to follow these steps:

Identify the given information and the unknown quantity.

Choose an appropriate sinusoidal model that fits the given information. You may need to use some trial and error or graphing technology to find the best fit.

Write an equation using the sinusoidal model and the given information. Substitute the values of the known quantities into the equation.

Solve the equation for the unknown quantity. You may need to use algebraic techniques, inverse functions, or calculators to find the solution.

Check your solution by plugging it back into the equation and verifying that it satisfies the given conditions.

## Examples and exercises

Here are some examples and exercises of using sinusoidal models in common core algebra II homework. Try to solve them on your own before looking at the solutions.

### Example 1

The height H (in feet) of a Ferris wheel car above the ground can be modeled by a sinusoidal function of time t (in minutes) as follows:

H = 50 \cdot \sin(\frac\pi3 \cdot t - \frac\pi2) + 55

Find the height of the car when t = 0, t = 1, and t = 2.

#### Solution

We can use the given equation to find the height of the car at any time by substituting the value of t into the equation and evaluating it. For example, when t = 0, we have:

H = 50 \cdot \sin(\frac\pi3 \cdot 0 - \frac\pi2) + 55

H = 50 \cdot \sin(-\frac\pi2) + 55

H = 50 \cdot (-1) + 55

H = -50 + 55

H = 5

So, the height of the car when t = 0 is 5 feet.

You can use a similar process to find the height of the car when t = 1 and t = 2. The answers are:

The height of the car when t = 1 is 80 feet.

The height of the car when t = 2 is 105 feet.

### Example 2

The temperature T (in degrees Fahrenheit) of a certain city on a spring day can be modeled by a sinusoidal function of time t (in hours) as follows:

T = 15 \cdot \cos(\frac\pi12 \cdot t - \frac\pi3) + 60

Find the highest and lowest temperatures of the day and when they occur.

#### Solution

We can use the given equation to find the highest and lowest temperatures of the day by finding the maximum and minimum points on the graph. To do this, we need to identify the values of a, b, c, and d from the equation. We have:

a = 15, which means the amplitude is 15 degrees.

b = \frac\pi12, which means the period is 24 hours.

c = -\frac\pi3, which means the phase shift is 4 hours to the right.

d = 60, which means the vertical shift is 60 degrees.

The maximum point occurs when x = -\fraccb, which is x = 4. The y-coordinate of the maximum point is y = a + d, which is y = 75. So, the highest temperature of the day is 75 degrees and it occurs at 4 a.m.

The minimum point occurs when x = -\fraccb + \fracT2, which is x = 16. The y-coordinate of the minimum point is y = -a + d, which is y = 45. So, the lowest temperature of the day is 45 degrees and it occurs at 4 p.m.

### Exercise 1

The length L (in feet) of daylight in a certain city on a given day of the year can be modeled by a sinusoidal function of time t (in days) as follows:

L = 4.8 \cdot \sin(\frac2\pi365 \cdot t - 1.7) + 12.2

Find the shortest and longest days of the year and how long they are.

### Exercise 2

The depth D (in feet) of water in a harbor can be modeled by a sinusoidal function of time t (in hours) as follows:

D = 8 \cdot \cos(\frac\pi6 \cdot t) + 12

Find the times when the depth of water is 10 feet and 20 feet.

### Exercise 3

The position P (in meters) of a point on a rotating wheel can be modeled by a sinusoidal function of time t (in seconds) as follows:

P = 1.5 \cdot \sin(4\pi \cdot t + \frac\pi2)

Find the period and amplitude of the function. Also, find the average position of the point over one period.

### Solution to exercise 1

We can use the given equation to find the shortest and longest days of the year by finding the maximum and minimum points on the graph. To do this, we need to identify the values of a, b, c, and d from the equation. We have:

a = 4.8, which means the amplitude is 4.8 feet.

b = \frac2\pi365, which means the period is 365 days.

c = -1.7, which means the phase shift is about 98 days to the right.

d = 12.2, which means the vertical shift is 12.2 feet.

The maximum point occurs when x = -\fraccb, which is x \approx 98. The y-coordinate of the maximum point is y = a + d, which is y = 17. So, the longest day of the year is 17 feet long and it occurs at day 98, which is around April 8.

The minimum point occurs when x = -\fraccb + \fracT2, which is x \approx 263. The y-coordinate of the minimum point is y = -a + d, which is y = 7.4. So, the shortest day of the year is 7.4 feet long and it occurs at day 263, which is around September 20.

### Solution to exercise 2

We can use the given equation to find the times when the depth of water is 10 feet and 20 feet by finding the x-intercepts and the maximum point on the graph. To do this, we need to identify the values of a, b, c, and d from the equation. We have:

a = 8, which means the amplitude is 8 feet.

b = \frac\pi6, which means the period is 12 hours.

c = 0, which means there is no phase shift.

d = 12, which means the vertical shift is 12 feet.

The x-intercepts occur when <cod

## Conclusion

In this article, we have learned how to use sinusoidal models to describe and analyze periodic phenomena. We have seen how to graph sinusoidal models, how to find their parameters, and how to use them in common core algebra II homework. We have also practiced some examples and exercises of sinusoidal modeling. We hope this article has helped you understand and appreciate the beauty and usefulness of sinusoidal models in mathematics and real life. d282676c82

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