To analyze and graph the function [tex]\( y = -3 \sin \left(\frac{1}{4} x\right) \)[/tex], let's identify the key characteristics of the function: the amplitude and the period.
### Amplitude
The amplitude of the sine function is defined as the absolute value of the coefficient in front of the sine term. Here, the function is [tex]\( y = -3 \sin \left(\frac{1}{4} x\right) \)[/tex]. The coefficient in front of the sine function is [tex]\(-3\)[/tex].
- The amplitude is [tex]\( \left| -3 \right| = 3 \)[/tex].
### Period
The period of the sine function is given by [tex]\( \frac{2\pi}{B} \)[/tex], where [tex]\( B \)[/tex] is the coefficient of [tex]\( x \)[/tex] inside the sine term.
Here, the function is [tex]\( y = -3 \sin \left(\frac{1}{4} x\right) \)[/tex].
- [tex]\( B \)[/tex] is [tex]\(\frac{1}{4}\)[/tex].
- The period is [tex]\( \frac{2\pi}{\frac{1}{4}} = 8\pi \)[/tex].
### Graph Features
Based on the amplitude and period:
- The amplitude of the graph is 3, meaning the function oscillates between [tex]\( y = -3 \)[/tex] and [tex]\( y = 3 \)[/tex].
- The period of the graph is [tex]\( 8\pi \)[/tex], so the function completes one full cycle every [tex]\( 8\pi \)[/tex] units along the [tex]\( x \)[/tex]-axis.
### Choosing the correct graph
Now, let's determine which graph matches these characteristics. We need the graph where:
1. The maximum value of [tex]\( y \)[/tex] is 3 and the minimum value is -3.
2. The function repeats every [tex]\( 8\pi \)[/tex] units.
Unfortunately, we do not have the images of the graphs labeled A, B, C, or D here, so we can't select the correct graph. However, the information provided should give you everything you need to identify the correct graph based on the amplitude and period.
### Summary
- Amplitude: [tex]\( 3 \)[/tex]
- Period: [tex]\( 8 \pi \)[/tex]
When choosing the correct graph, look for the one that matches these characteristics.
