Certainly! Let’s go through each part step-by-step.
### Part (a): Sine Function
- Amplitude: The amplitude of a sine function refers to the maximum distance from the midline to the peak or trough of the wave. Given that the amplitude is 2, this changes the equation from [tex]\( y = \sin(x) \)[/tex] to [tex]\( y = 2\sin(x) \)[/tex].
- Period: The period of the sine function is given as [tex]\( 2\pi \)[/tex]. The standard sine function [tex]\( y = \sin(x) \)[/tex] already has a period of [tex]\( 2\pi \)[/tex]. Therefore, no adjustment is needed for the period.
- Midline: The midline is an additional vertical shift in the function. If the midline is [tex]\( y = -5 \)[/tex], this effectively moves the whole wave down by 5 units. Thus, we adjust the equation to include this vertical shift: [tex]\( y = 2\sin(x) - 5 \)[/tex].
Putting all these together, the equation for the sine function is:
[tex]\[ y = 2\sin(x) - 5 \][/tex]
### Part (b): Cosine Function
- Phase Shift: A phase shift involves a horizontal shift of the graph. The given phase shift is [tex]\( \frac{\pi}{3} \)[/tex] units to the left. To shift a cosine function [tex]\( \frac{\pi}{3} \)[/tex] units to the left, we replace [tex]\( x \)[/tex] with [tex]\( x + \frac{\pi}{3} \)[/tex] in the equation. This changes the equation from [tex]\( y = \cos(x) \)[/tex] to [tex]\( y = \cos(x + \frac{\pi}{3}) \)[/tex].
- Period: The period of the cosine function is provided as [tex]\( 2\pi \)[/tex]. The standard cosine function [tex]\( y = \cos(x) \)[/tex] already has a period of [tex]\( 2\pi \)[/tex], so no further adjustment is needed for the period.
Therefore, the equation for the cosine function is:
[tex]\[ y = \cos\left(x + \frac{\pi}{3}\right) \][/tex]
### Summary
So, the equations you need are:
1. [tex]\( y = 2\sin(x) - 5 \)[/tex] for the sine function with the specified properties.
2. [tex]\( y = \cos\left(x + \frac{\pi}{3}\right) \)[/tex] for the cosine function with the specified properties.
These equations incorporate the given amplitude, period, midline shift, and phase shift exactly as required.
