Answered

Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Join our Q&A platform to get precise answers from experts in diverse fields and enhance your understanding. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

Write an equation for each of the following:

a. A sine function with a period of [tex]\( 2\pi \)[/tex], an amplitude of 2, and a midline at [tex]\( y = -5 \)[/tex]:
[tex]\[ y = 2 \sin(x) - 5 \][/tex]

b. A cosine function with a phase shift of [tex]\( \frac{\pi}{3} \)[/tex] units to the left:
[tex]\[ y = \cos\left(x + \frac{\pi}{3}\right) \][/tex]

Sagot :

GinnyAnswer
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.
We hope you found this helpful. Feel free to come back anytime for more accurate answers and updated information. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.