Before we delve into graphing the specific function f(x) = 3*sin(x), let's first understand the fundamental characteristics of the sine function. The sine function, denoted as sin(x), is a trigonometric function that represents the ratio of the opposite side to the hypotenuse in a right triangle. It exhibits a periodic...
The sine function has a period of 2π, which means its graph completes one full cycle every 2π units along the x-axis. Its amplitude is 1, indicating the maximum and minimum values of the function are 1 and -1 respectively. The graph oscillates between these values, starting at the origin (0, 0) and moving upwards to its peak at (π/2, 1), then descending to its trough at (π, -1), and finally returning to the origin at (2π, 0).
Understanding the Transformation
The function f(x) = 3*sin(x) represents a transformation of the basic sine function. The multiplication by 3 in front of the sine function affects the amplitude of the graph.
Amplitude and its Impact on the Graph
The amplitude of the sine function represents the distance from the center line to the maximum or minimum point of the graph. In the original sine function, sin(x), the amplitude is 1. However, in f(x) = 3*sin(x), the amplitude is multiplied by 3, resulting in an amplitude of 3. This means the graph will now oscillate between 3 and -3 instead of 1 and -1. The vertical stretch by a factor of 3 expands the graph vertically, making it taller.
Graphing the Function
To graph f(x) = 3*sin(x), we can use the following steps:
- Identify the Amplitude: As we established earlier, the amplitude is 3.
- Identify the Period: The period of the sine function is 2π. Since there are no other transformations affecting the period, the period of f(x) = 3*sin(x) remains 2π.
- Plot Key Points: Start by plotting the key points of the sine function within one period (0 to 2π):
- (0, 0)
- (π/2, 3)
- (π, 0)
- (3π/2, -3)
- (2π, 0)
- Connect the Points: Connect the points smoothly, forming a wave-like curve. The curve will oscillate between 3 and -3, completing one full cycle every 2π units along the x-axis.
- Extend the Graph: Since the sine function is periodic, you can repeat the same pattern of the graph to the left and right of the initial period (0 to 2π) to represent the function's infinite repetition.
Visual Representation
The graph of f(x) = 3*sin(x) will look like a standard sine wave but with a taller amplitude, reaching a maximum of 3 and a minimum of -3.
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3 | *********
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-3 | *******
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Key Takeaways
In conclusion, graphing f(x) = 3*sin(x) involves understanding the transformations applied to the basic sine function. The multiplication by 3 in front of sin(x) changes the amplitude, making the graph taller and oscillate between 3 and -3. The period remains unchanged at 2π. By following the steps outlined above, you can accurately graph f(x) = 3*sin(x) and visualize its periodic behavior.