Question

Write a cosine function that passes through the points (1/2,1) and (3/2,-3)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the equation of the cosine function that passes through the points (1/2,1) and (3/2,-3), we first need to determine the amplitude, period, phase shift, and vertical shift of the function.

Given that the function is a cosine function, we can use the general formula:

y = A cos(Bx - C) + D

where:
A = amplitude
B = 2π / period
C = phase shift
D = vertical shift

Now, let's start by finding the amplitude (A). The amplitude is the distance between the maximum and minimum points of the function. Since the points given are (1/2,1) and (3/2,-3), the amplitude is |1 - (-3)| = 4.

Next, we determine the period of the function. The period is the length of one cycle of the cosine function. Since the function passes through two points that are one complete cycle apart, the period is 2.

Now, let's find the phase shift. The phase shift is the horizontal shift of the cosine function. In this case, the function is not shifted horizontally, so the phase shift is 0.

Lastly, we determine the vertical shift. The vertical shift is the vertical translation of the function. The average of the maximum and minimum of the function will give us the vertical shift.

Average = (1 + (-3)) / 2 = -1

Now, we can substitute the values of A, B, C, and D into the general equation to get the function:

y = 4 cos(2πx/2 - 0) - 1
= 4 cos(πx) - 1

Therefore, the cosine function that passes through the points (1/2,1) and (3/2,-3) is y = 4 cos(πx) - 1.