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Introduction

In mathematics, trigonometric functions are a crucial part of algebra and calculus. These functions describe the relationships between the angles and side lengths of triangles. One of the most common trigonometric functions is the sine function, which is represented by the symbol sin(x). In this article, we will explore the key features of the function f(x) = (5/4) sin(x) + 1, and how to determine its maximum and minimum values.

The Graph of the Function

The graph of the function f(x) = (5/4) sin(x) + 1 is a sinusoidal curve that oscillates between two extreme values. The graph is a result of the sine function being multiplied by a constant factor of 5/4 and then shifted upwards by 1 unit.

Key Features of the Function

The key features of the function f(x) = (5/4) sin(x) + 1 include:

• Amplitude: The amplitude of the function is the distance from the midline to the maximum or minimum value of the function. In this case, the amplitude is 5/4, which means that the function oscillates between 1 + 5/4 = 9/4 and 1 - 5/4 = -1/4.
• Period: The period of the function is the distance between two consecutive points on the graph that have the same y-coordinate. The period of the sine function is 2π, but in this case, the period is still 2π because the constant factor of 5/4 does not affect the period.
• Midline: The midline of the function is the horizontal line that passes through the midpoint of the function. In this case, the midline is y = 1.
• Maximum and Minimum Values: The maximum value of the function is the highest point on the graph, while the minimum value is the lowest point on the graph. We will discuss how to determine these values in the next section.

Determining the Maximum and Minimum Values

To determine the maximum and minimum values of the function f(x) = (5/4) sin(x) + 1, we need to find the values of x that correspond to the maximum and minimum values of the sine function.

• Maximum Value: The maximum value of the sine function occurs when sin(x) = 1. Therefore, the maximum value of the function f(x) = (5/4) sin(x) + 1 occurs when (5/4) sin(x) + 1 = 9/4, which gives sin(x) = 1. This means that the maximum value of the function is 9/4.
• Minimum Value: The minimum value of the sine function occurs when sin(x) = -1. Therefore, the minimum value of the function f(x) = (5/4) sin(x) + 1 occurs when (5/4) sin(x) + 1 = -1/4, which gives sin(x) = -1. This means that the minimum value of the function is -1/4.

Conclusion

In conclusion, the key features of the function f(x) = (5/4) sin(x) + 1 include an amplitude of 5/4, a period of 2π, a midline of y = 1, and maximum and minimum values of 9/4 and -1/4, respectively. By understanding these key features, we can better analyze and interpret the behavior of the function.

Key Takeaways

• The amplitude of the function f(x) = (5/4) sin(x) + 1 is 5/4.
• The period of the function is 2π.
• The midline of the function is y = 1.
• The maximum value of the function is 9/4.
• The minimum value of the function is -1/4.

Final Thoughts

Q: What is the amplitude of the function f(x) = (5/4) sin(x) + 1?

A: The amplitude of the function f(x) = (5/4) sin(x) + 1 is 5/4. This means that the function oscillates between 1 + 5/4 = 9/4 and 1 - 5/4 = -1/4.

Q: What is the period of the function f(x) = (5/4) sin(x) + 1?

A: The period of the function f(x) = (5/4) sin(x) + 1 is 2π. This is because the constant factor of 5/4 does not affect the period of the sine function.

Q: What is the midline of the function f(x) = (5/4) sin(x) + 1?

A: The midline of the function f(x) = (5/4) sin(x) + 1 is y = 1. This is because the function is shifted upwards by 1 unit.

Q: What are the maximum and minimum values of the function f(x) = (5/4) sin(x) + 1?

A: The maximum value of the function f(x) = (5/4) sin(x) + 1 is 9/4, which occurs when sin(x) = 1. The minimum value of the function is -1/4, which occurs when sin(x) = -1.

Q: How do I determine the maximum and minimum values of the function f(x) = (5/4) sin(x) + 1?

A: To determine the maximum and minimum values of the function f(x) = (5/4) sin(x) + 1, you need to find the values of x that correspond to the maximum and minimum values of the sine function. This can be done by setting sin(x) = 1 and sin(x) = -1, respectively.

Q: What is the significance of the function f(x) = (5/4) sin(x) + 1 in real-world applications?

A: The function f(x) = (5/4) sin(x) + 1 is a simple example of a sinusoidal function, which is used to model many real-world phenomena, such as sound waves, light waves, and population growth.

Q: Can I use the function f(x) = (5/4) sin(x) + 1 to model a specific real-world problem?

A: Yes, you can use the function f(x) = (5/4) sin(x) + 1 to model a specific real-world problem, such as the motion of a pendulum or the growth of a population. However, you would need to adjust the function to fit the specific problem you are trying to model.

Q: How do I graph the function f(x) = (5/4) sin(x) + 1?

A: To graph the function f(x) = (5/4) sin(x) + 1, you can use a graphing calculator or a computer program. You can also use a table of values to plot the function.

Q: Can I use the function f(x) = (5/4) sin(x) + 1 to solve a system of equations?

A: No, the function f(x) = (5/4) sin(x) + 1 is a single function and cannot be used to solve a system of equations. However, you can use the function to solve a single equation or to model a real-world problem.

Q: How do I find the derivative of the function f(x) = (5/4) sin(x) + 1?

A: To find the derivative of the function f(x) = (5/4) sin(x) + 1, you can use the chain rule and the fact that the derivative of sin(x) is cos(x). The derivative of the function is f'(x) = (5/4) cos(x).