The diagram shows the graph of y=sin x

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 ):

  1. 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.
  2. 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).
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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