To properly graph one cycle of the trigonometric function [tex]\( y = 7 \cos 6x \)[/tex], follow these step-by-step instructions:
1. Determine the Period: The period of a cosine function [tex]\( y = a \cos(bx) \)[/tex] is given by [tex]\( \frac{2\pi}{b} \)[/tex]. Here, [tex]\( b = 6 \)[/tex], so the period is:
[tex]\[
\text{Period} = \frac{2\pi}{6} = \frac{\pi}{3}
\][/tex]
2. Determine the Amplitude: The amplitude is given by the value of [tex]\( a \)[/tex] in the cosine function. Here, [tex]\( a = 7 \)[/tex], so the amplitude is 7.
3. Set the X-axis Labels: To graph exactly one cycle, we need to plot the function from [tex]\( x = 0 \)[/tex] to [tex]\( x = \frac{\pi}{3} \)[/tex]. Therefore, the x-axis labels should start from 0 and go up to [tex]\( \frac{\pi}{3} \)[/tex].
4. Set the Y-axis Labels: The amplitude of the function [tex]\( y = 7 \cos 6x \)[/tex] ranges from -7 to 7. Therefore, the y-axis should be labeled from -7 to 7.
5. Plot Key Points: The cosine function completes one cycle in this period:
- At [tex]\( x = 0 \)[/tex], [tex]\( y = 7 \cos(6 \cdot 0) = 7 \cos(0) = 7 \)[/tex].
- At [tex]\( x = \frac{\pi}{6} \)[/tex] (a quarter period), [tex]\( y = 7 \cos\left(6 \cdot \frac{\pi}{6}\right) = 7 \cos(\pi) = -7 \)[/tex].
- At [tex]\( x = \frac{\pi}{3} \)[/tex] (half period), [tex]\( y = 7 \cos\left(6 \cdot \frac{\pi}{3}\right) = 7 \cos(2\pi) = 7 \)[/tex].
6. Graph the Function: Plot the points and sketch the curve:
- Start at (0, 7).
- Decrease smoothly to the minimum point at [tex]\( x = \frac{\pi}{6}, y = -7 \)[/tex].
- Return back to the point ( [tex]\( x = \frac{\pi}{3}, y = 7 \)[/tex] ).
7. Label the Axes:
- The x-axis should be labeled with the key points: 0, [tex]\( \frac{\pi}{6} \)[/tex], and [tex]\( \frac{\pi}{3} \)[/tex].
- The y-axis should be labeled from -7 to 7.
So, the x-axis label would be:
- 0
- [tex]\( \frac{\pi}{6} \)[/tex]
- [tex]\( \frac{\pi}{3} \)[/tex]
And for the y-axis label:
- -7
- 7
Graphing tools or graph paper will help in making the graph precise, with smooth transitions between these key points.
