Sine Graphs 4Dbc4B

1. **State the problem:** We want to sketch the graphs of $y=\sin x$ and $y=\sin 2x$ and understand the basic rules for these sine functions. 2. **Recall the basic sine function:** The function $y=\sin x$ has the following properties: - Amplitude: 1 (maximum value 1, minimum value -1) - Period: $2\pi$ (the length of one full cycle) - Zeroes: at multiples of $\pi$ (i.e., $x=0, \pi, 2\pi, \ldots$) - Shape: smooth wave oscillating between -1 and 1 3. **Effect of the coefficient inside the sine argument:** For $y=\sin 2x$, the coefficient 2 affects the period. - New period = $\frac{2\pi}{2} = \pi$ - This means the sine wave completes one full cycle twice as fast as $y=\sin x$ 4. **Basic rules for sketching:** - Identify amplitude (coefficient before sine, here 1 for both) - Identify period using formula $\text{Period} = \frac{2\pi}{|b|}$ where $b$ is the coefficient of $x$ - Plot key points: zeros, maxima, minima within one period - Repeat the pattern for other intervals 5. **Intermediate work:** - For $y=\sin x$, period is $2\pi$ - For $y=\sin 2x$, period is $\pi$ 6. **Summary:** - $y=\sin x$ oscillates between -1 and 1 with period $2\pi$ - $y=\sin 2x$ oscillates between -1 and 1 with period $\pi$ - The graph of $y=\sin 2x$ is a horizontally compressed version of $y=\sin x$ by a factor of 2 This completes the explanation and basic rules for sketching these sine graphs.