Questions: (a) The sine and cosine curves y=a sin (k x) and y=a cos (k x), k>0 have amplitude and period. The sine curve y=5 sin (2 x) has amplitude, period π, an appropriate interval to graph one period is (0, ). (b) Graph one period of y=5 sin (2 x) and label the x-coordinates of the key points used for graphing the function.

(a) The sine and cosine curves y=a sin (k x) and y=a cos (k x), k>0 have amplitude and period. The sine curve y=5 sin (2 x) has amplitude, period π, an appropriate interval to graph one period is (0, ). (b) Graph one period of y=5 sin (2 x) and label the x-coordinates of the key points used for graphing the function.
Transcript text: (a) The sine and cosine curves $y=a \sin (k x)$ and $y=a \cos (k x), k>0$ have amplitude $\square$ and period $\square$ - The sine curve $y=5 \sin (2 x)$ has amplitude $\square$ period $\pi$ an appropriate interval to graph one period is $(0$, $\square$ ). (b) Graph one period of $y=5 \sin (2 x)$ and label the $x$-coordinates of the key points used for graphing the function.
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Solution

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Solution Steps

Step 1: Identify the amplitude and period of the sine and cosine curves
  • The sine and cosine curves \( y = a \sin(kx) \) and \( y = a \cos(kx) \) have amplitude \( a \) and period \( \frac{2\pi}{k} \).
Step 2: Determine the amplitude and period of the given sine curve
  • The sine curve \( y = 5 \sin(2x) \) has amplitude \( 5 \) and period \( \frac{2\pi}{2} = \pi \).
Step 3: Identify the appropriate interval to graph one period
  • An appropriate interval to graph one period of \( y = 5 \sin(2x) \) is \( [0, \pi] \).

Final Answer

  • Amplitude: \( 5 \)
  • Period: \( \pi \)
  • Appropriate interval: \( [0, \pi] \)
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