Understanding the General Form of a Sine Function

Oct 13, 2025 by miodeya.com 108 views

Identifying Key Features of the Graph of $y = -4 \sin (2x) + 2$

The graph of a sine function is a fundamental concept in mathematics, and understanding its key features is crucial for solving various problems in trigonometry and beyond. In this article, we will delve into the graph of $y = -4 \sin (2x) + 2$ and identify its amplitude, period, and shift up units.

Understanding the General Form of a Sine Function

Before we dive into the specifics of the given function, let's first understand the general form of a sine function. The general form of a sine function is given by:

y = A \sin (Bx) + C

where $A$ is the amplitude, $B$ is the frequency, and $C$ is the vertical shift.

Identifying the Amplitude

The amplitude of a sine function is the maximum value that the function attains. In the given function, $y = -4 \sin (2x) + 2$ , the amplitude is $|A| = |-4| = 4$ . This means that the graph of the function will oscillate between $y = -4$ and $y = 2$ .

Identifying the Period

The period of a sine function is the distance between two consecutive points on the graph that have the same $y$ -coordinate. The period of a sine function is given by:

\text{Period} = \frac{2\pi}{|B|}

In the given function, $y = -4 \sin (2x) + 2$ , the frequency is $|B| = |2| = 2$ . Therefore, the period of the function is:

\text{Period} = \frac{2\pi}{|2|} = \pi

This means that the graph of the function will complete one full cycle in an interval of length $\pi$ .

Identifying the Shift Up Units

The shift up units of a sine function is the vertical shift of the graph. In the given function, $y = -4 \sin (2x) + 2$ , the vertical shift is $C = 2$ . This means that the graph of the function will be shifted up by 2 units.

Key Points of the Graph

To find the key points of the graph, we need to find the points where the function attains its maximum and minimum values. The maximum value of the function is attained when $\sin (2x) = 1$ , and the minimum value is attained when $\sin (2x) = -1$ . Therefore, the key points of the graph are:

$(0, -2)$
$(\frac{\pi}{4}, 2)$
$(\frac{\pi}{2}, 0)$
$(\frac{3\pi}{4}, -2)$
$(\pi, -4)$

Conclusion

In conclusion, the graph of $y = -4 \sin (2x) + 2$ has an amplitude of 4, a period of $\pi$ , and a shift up units of 2. The key points of the graph are $(0, -2)$ , $(\frac{\pi}{4}, 2)$ , $(\frac{\pi}{2}, 0)$ , $(\frac{3\pi}{4}, -2)$ , and $(\pi, -4)$ . Understanding the key features of the graph of a sine function is crucial for solving various problems in trigonometry and beyond.

References

[1] "Trigonometry" by Michael Corral
[2] "Calculus" by Michael Spivak

Discussion

What are some other key features of the graph of a sine function? How do you determine the amplitude, period, and shift up units of a sine function? What are some real-world applications of the graph of a sine function?
Q&A: Identifying Key Features of the Graph of $y = -4 \sin (2x) + 2$

In our previous article, we discussed the key features of the graph of $y = -4 \sin (2x) + 2$ , including its amplitude, period, and shift up units. In this article, we will answer some frequently asked questions about the graph of a sine function.

Q: What is the amplitude of the graph of $y = -4 \sin (2x) + 2$ ?

A: The amplitude of the graph of $y = -4 \sin (2x) + 2$ is 4. This means that the graph of the function will oscillate between $y = -4$ and $y = 2$ .

Q: What is the period of the graph of $y = -4 \sin (2x) + 2$ ?

A: The period of the graph of $y = -4 \sin (2x) + 2$ is $\pi$ . This means that the graph of the function will complete one full cycle in an interval of length $\pi$ .

Q: What is the shift up units of the graph of $y = -4 \sin (2x) + 2$ ?

A: The shift up units of the graph of $y = -4 \sin (2x) + 2$ is 2. This means that the graph of the function will be shifted up by 2 units.

Q: How do you determine the amplitude, period, and shift up units of a sine function?

A: To determine the amplitude, period, and shift up units of a sine function, you need to look at the general form of the function, which is given by:

y = A \sin (Bx) + C

The amplitude is given by $|A|$ , the period is given by $\frac{2\pi}{|B|}$ , and the shift up units is given by $C$ .

Q: What are some real-world applications of the graph of a sine function?

A: The graph of a sine function has many real-world applications, including:

Modeling the motion of a pendulum
Modeling the motion of a wave
Modeling the temperature of a room
Modeling the population of a species

Q: How do you find the key points of the graph of a sine function?

A: To find the key points of the graph of a sine function, you need to find the points where the function attains its maximum and minimum values. The maximum value of the function is attained when $\sin (2x) = 1$ , and the minimum value is attained when $\sin (2x) = -1$ . Therefore, the key points of the graph are:

$(0, -2)$
$(\frac{\pi}{4}, 2)$
$(\frac{\pi}{2}, 0)$
$(\frac{3\pi}{4}, -2)$
$(\pi, -4)$

Q: What are some other key features of the graph of a sine function?

A: Some other key features of the graph of a sine function include:

The x-intercepts, which are the points where the graph intersects the x-axis
The y-intercept, which is the point where the graph intersects the y-axis
The asymptotes, which are the lines that the graph approaches as x approaches infinity or negative infinity

Conclusion

References

[1] "Trigonometry" by Michael Corral
[2] "Calculus" by Michael Spivak

Discussion