Pregunta
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Determine the amplitude, period, and phase
shift (if any) of the given function. Graph the
function.

The amplitude is

Ask by Ryan Hilton. in the United States
Dec 08,2024

Solución de inteligencia artificial de Upstudy

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Amplitude: 2, Period: , Phase Shift: 0

Solución

To analyze the function , we can determine the amplitude, period, and phase shift as follows:
  1. Amplitude: The amplitude of a sine function is given by the absolute value of . In this case, , so the amplitude is:
  2. Period: The period of a sine function is given by the formula . In this case, (since there is no coefficient in front of ), so the period is:
  3. Phase Shift: The phase shift is determined by the term in the function . In this case, there is no term (it is effectively 0), so the phase shift is:

Summary:

  • Amplitude:
  • Period:
  • Phase Shift:

Graphing the Function:

To graph :
  • The sine function normally oscillates between -1 and 1. However, since we have a coefficient of -2, the function will oscillate between -2 and 2, but inverted (i.e., it will reach its maximum at -0 and minimum at -2).
  • The function will complete one full cycle over the interval .
Key Points to Plot:
  • At ,
  • At ,
  • At ,
  • At ,
  • At ,

Graph:

The graph will start at , go down to , return to , rise to , and return to .
You can sketch the graph based on these points, ensuring to reflect the sine wave’s typical shape but inverted and scaled vertically.

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The Deep Dive

The amplitude of the function is given by the absolute value of the coefficient in front of the sine function. Here, the coefficient is -2, resulting in an amplitude of .
The period of the sine function is typically . Since there are no modifications to the variable (like a coefficient that would stretch or compress), the period remains .
Finally, there is no phase shift in this equation, as there are no additions or subtractions inside the sine function to adjust its position along the x-axis.
In summary:
  • Amplitude: 2
  • Period:
  • Phase shift: 0
To visualize this function, you would graph the sine wave reflected across the x-axis (due to the negative sign) but maintaining the amplitude of 2. It would oscillate between 2 and -2 over each interval of .

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