Understanding the Graph of a Sinusoidal Function

Oct 13, 2025 by pynkocean.com 23 views

Introduction

In mathematics, sinusoidal functions are a fundamental concept in trigonometry and are used to model various real-world phenomena, such as sound waves, light waves, and population growth. These functions have a characteristic graph that oscillates between a maximum and minimum value, with a midline that represents the average value of the function. In this article, we will explore the graph of a sinusoidal function that intersects its midline at a specific point and has a minimum point at another point. We will use this information to write the formula of the function.

Understanding the Graph of a Sinusoidal Function

A sinusoidal function is a function that can be written in the form:

f(x) = a sin(bx + c) + d

where:

a is the amplitude of the function, which represents the maximum value of the function.
b is the frequency of the function, which represents the number of oscillations per unit of x.
c is the phase shift of the function, which represents the horizontal shift of the function.
d is the vertical shift of the function, which represents the midline of the function.

The graph of a sinusoidal function is a periodic curve that oscillates between a maximum and minimum value. The midline of the function is the horizontal line that represents the average value of the function.

Intersections with the Midline

The problem states that the graph of the sinusoidal function intersects its midline at the point (0, -7). This means that the function has a value of -7 at x = 0. We can use this information to find the value of d, which is the vertical shift of the function.

Since the function intersects its midline at (0, -7), we know that:

f(0) = -7

Substituting x = 0 into the equation of the function, we get:

-7 = a sin(c) + d

Since the function intersects its midline at (0, -7), we know that the value of d is equal to -7. Therefore, we can write:

d = -7

Minimum Point

The problem states that the graph of the sinusoidal function has a minimum point at the point (π/4, -9). This means that the function has a value of -9 at x = π/4. We can use this information to find the value of a, which is the amplitude of the function.

Since the function has a minimum point at (π/4, -9), we know that:

f(π/4) = -9

Substituting x = π/4 into the equation of the function, we get:

-9 = a sin(bπ/4 + c) - 7

Simplifying the equation, we get:

-9 + 7 = a sin(bπ/4 + c)

-2 = a sin(bπ/4 + c)

Since the function has a minimum point at (π/4, -9), we know that the value of a is equal to 2. Therefore, we can write:

a = 2

Writing the Formula of the Function

Now that we have found the values of a, d, and the coordinates of the minimum point, we can write the formula of the function.

Since the function has a minimum point at (π/4, -9), we know that the value of b is equal to 1. Therefore, we can write:

b = 1

Substituting the values of a, b, and d into the equation of the function, we get:

f(x) = 2 sin(x + c) - 7

To find the value of c, we can use the fact that the function has a minimum point at (π/4, -9). Substituting x = π/4 into the equation of the function, we get:

-9 = 2 sin(π/4 + c) - 7

Simplifying the equation, we get:

-9 + 7 = 2 sin(π/4 + c)

-2 = 2 sin(π/4 + c)

Dividing both sides by 2, we get:

-1 = sin(π/4 + c)

Since the sine function is equal to -1 at π/2, we know that:

π/4 + c = 3π/2

Subtracting π/4 from both sides, we get:

c = 3π/2 - π/4

c = 5π/4

Substituting the value of c into the equation of the function, we get:

f(x) = 2 sin(x + 5π/4) - 7

Conclusion

In this article, we have used the information about the intersections and minimum points of a sinusoidal function to write the formula of the function. We have found the values of a, b, c, and d, and have used these values to write the equation of the function. The formula of the function is:

f(x) = 2 sin(x + 5π/4) - 7

This formula represents a sinusoidal function that intersects its midline at (0, -7) and has a minimum point at (π/4, -9).

Introduction

In our previous article, we explored the graph of a sinusoidal function that intersects its midline at a specific point and has a minimum point at another point. We used this information to write the formula of the function. In this article, we will answer some common questions related to the graph of a sinusoidal function.

Q: What is the amplitude of the function?

A: The amplitude of the function is the maximum value of the function. In our previous article, we found that the amplitude of the function is 2.

Q: What is the frequency of the function?

A: The frequency of the function is the number of oscillations per unit of x. In our previous article, we found that the frequency of the function is 1.

Q: What is the phase shift of the function?

A: The phase shift of the function is the horizontal shift of the function. In our previous article, we found that the phase shift of the function is 5π/4.

Q: What is the vertical shift of the function?

A: The vertical shift of the function is the midline of the function. In our previous article, we found that the vertical shift of the function is -7.

Q: How do I find the equation of a sinusoidal function?

A: To find the equation of a sinusoidal function, you need to know the amplitude, frequency, phase shift, and vertical shift of the function. You can use this information to write the equation of the function in the form:

f(x) = a sin(bx + c) + d

Q: What is the significance of the midline of a sinusoidal function?

A: The midline of a sinusoidal function represents the average value of the function. It is the horizontal line that the function oscillates around.

Q: How do I find the minimum point of a sinusoidal function?

A: To find the minimum point of a sinusoidal function, you need to know the amplitude, frequency, and phase shift of the function. You can use this information to find the x-coordinate of the minimum point, and then substitute this value into the equation of the function to find the y-coordinate.

Q: Can a sinusoidal function have a maximum point?

A: Yes, a sinusoidal function can have a maximum point. The maximum point is the point on the graph where the function has its maximum value.

Q: How do I graph a sinusoidal function?

A: To graph a sinusoidal function, you can use a graphing calculator or a computer program to plot the function. You can also use a table of values to plot the function.

Q: Can a sinusoidal function be used to model real-world phenomena?

A: Yes, a sinusoidal function can be used to model real-world phenomena such as sound waves, light waves, and population growth.

Q: How do I find the equation of a sinusoidal function that models a real-world phenomenon?

A: To find the equation of a sinusoidal function that models a real-world phenomenon, you need to collect data on the phenomenon and use this data to find the equation of the function.

Conclusion

In this article, we have answered some common questions related to the graph of a sinusoidal function. We have discussed the amplitude, frequency, phase shift, and vertical shift of the function, and have provided examples of how to find the equation of a sinusoidal function. We have also discussed the significance of the midline of a sinusoidal function and how to find the minimum point of a sinusoidal function.