the diagram shows the graph of y=sin x
The Diagram of the Graph of ( y = \sin x )
The function ( y = \sin x ) represents one of the most important trigonometric functions in mathematics. It describes a periodic, wave-like behavior and is widely used in fields such as physics, engineering, and signal processing. Below, I’ll explain the features, behaviors, and properties of the graph of ( y = \sin x ), as well as provide examples and analysis to help you understand its structure.
Key Features of ( y = \sin x ):
-
Periodic Nature:
- The function ( y = \sin x ) is periodic, meaning it repeats itself at regular intervals.
- The period of ( y = \sin x ) is ( 2\pi ), which means the graph repeats every ( 2\pi ) units along the ( x )-axis.
-
Amplitude:
- The amplitude represents the maximum displacement from the horizontal axis.
- For ( y = \sin x ), the amplitude is 1, since the graph oscillates between 1 (the maximum value) and -1 (the minimum value).
-
Zeroes:
- The function ( y = \sin x ) intersects the ( x )-axis (i.e., ( y = 0 )) at regular intervals.
- The zeroes of ( y = \sin x ) are located at ( x = n\pi ), where ( n ) is any integer.
-
Maximum and Minimum Points:
- The graph reaches its maximum value, ( +1 ), at ( x = \frac{\pi}{2} + 2n\pi ), where ( n ) is any integer.
- The graph reaches its minimum value, ( -1 ), at ( x = \frac{3\pi}{2} + 2n\pi ), where ( n ) is any integer.
-
Shape:
- The graph of ( y = \sin x ) starts at 0 (when ( x = 0 )), rises to +1, decreases to -1, and then returns to 0 to complete one cycle. This wave-like shape is symmetric about the origin.
-
Domain and Range:
- The domain of ( y = \sin x ) is all real numbers (( x \in \mathbb{R} )), as the sine function is defined for all values of ( x ).
- The range of ( y = \sin x ) is ( -1 \leq y \leq 1 ), because the sine function cannot exceed these bounds.
-
Wave-Like Behavior:
- The sine graph oscillates in both positive and negative directions, resembling a smooth, continuous wave.
Equation of ( y = \sin x ): The General Case
The general form of the sine function is:
[
y = A \sin(Bx + C) + D
]
Where:
- ( A ): Amplitude, which defines the vertical stretch (default is 1).
- ( B ): Frequency, which adjusts the number of oscillations per unit of ( x ) (default is 1).
- ( C ): Phase Shift, which moves the graph left or right.
- ( D ): Vertical Shift, which moves the graph up or down.
When ( A = 1 ), ( B = 1 ), ( C = 0 ), and ( D = 0 ), the equation reduces to ( y = \sin x ), the standard sine function.
The Graph of ( y = \sin x ):
To visualize the behavior of ( y = \sin x ), here are some important points and how they relate to the graph:
| ( x ) (radians) | ( y = \sin x ) | Key Points on Graph |
|---|---|---|
| ( 0 ) | ( 0 ) | Zero at the origin |
| ( \frac{\pi}{2} ) | ( 1 ) | Maximum value |
| ( \pi ) | ( 0 ) | Zero |
| ( \frac{3\pi}{2} ) | ( -1 ) | Minimum value |
| ( 2\pi ) | ( 0 ) | Zero (one cycle) |
Using these points, you can sketch one cycle of the sine wave on the interval ( [0, 2\pi] ). After that, the cycle repeats in both the positive and negative directions on the ( x )-axis.
Characteristics of ( y = \sin x )
1. Behavior on Intervals:
- On ( 0 \leq x \leq \pi ), ( y = \sin x ) increases from 0 to 1, then decreases back to 0.
- On ( \pi \leq x \leq 2\pi ), ( y = \sin x ) decreases to -1, then increases back to 0.
2. Symmetry:
- The function ( y = \sin x ) is odd, meaning it has origin symmetry:
- ( \sin(-x) = -\sin(x) )
3. Continuity and Smoothness:
- The graph of ( y = \sin x ) is continuous (no gaps or breaks) and smooth (no sharp corners).
Graph Example and Markdown-Diagram:
Here’s how you can visualize the graph of ( y = \sin x ). On a coordinate plane:
- Draw the sine wave starting from ( (0, 0) ), rising to ( ( \frac{\pi}{2}, 1 ) ), returning to ( (\pi, 0) ), dipping to ( ( \frac{3\pi}{2}, -1 ) ), and back at ( (2\pi, 0) ).
Amplitude: +1 and -1 y
__________
+1 | /| || +1
| / | ||
| / | ||
0 + -------------------------...----+--------------------> x
-1 | |-Pi|
| Shift would Fundo y-Clause