Questions: Graph two full periods of the function. y=5 sin (5x+20)-1

Graph two full periods of the function. y=5 sin (5x+20)-1
Transcript text: Graph two full periods of the function. \[ y=5 \sin (5 x+20)-1 \]
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Solution

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Solution Steps

Step 1: Determine the period of the function

The period of the sine function \( y = A \sin(Bx + C) + D \) is given by \( \frac{2\pi}{|B|} \). For the function \( y = 5 \sin(5x + 20) - 1 \), \( B = 5 \).

\[ \text{Period} = \frac{2\pi}{5} \]

Step 2: Calculate the range for two full periods

Two full periods would be:

\[ 2 \times \frac{2\pi}{5} = \frac{4\pi}{5} \]

Step 3: Determine the x-axis range

To graph two full periods, we need to find the x-axis range. We can start from \( x = 0 \) to \( x = \frac{4\pi}{5} \).

\[ \text{x-axis range} = (0, \frac{4\pi}{5}) \]

Step 4: Determine the y-axis range

The function \( y = 5 \sin(5x + 20) - 1 \) has an amplitude of 5 and is shifted down by 1. Therefore, the range of \( y \) is:

\[ [-6, 4] \]

Final Answer

The x-axis range is from \( 0 \) to \( \frac{4\pi}{5} \) and the y-axis range is from \( -6 \) to \( 4 \).

{"axisType": 3, "coordSystem": {"xmin": 0, "xmax": 2.51328, "ymin": -6, "ymax": 4}, "commands": ["y = 5sin(5x + 20) - 1"], "latex_expressions": ["$y = 5 \\sin(5x + 20) - 1$"]}

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