Question

What is the cosine equation of the function shown?​

Answer 1

Answer: f(x)=5cos(x+pi/3)+4

Step-by-step explanation: I just tool the quiz

Answer 2

The cosine equation is

`\[ f(x) = 6\cos(x - \pi) + 2 \]`

To find the cosine equation of the function shown in the image, we follow these steps:

1. Identify the Amplitude: The amplitude of a trigonometric function is the distance from the midline of the graph to a peak or trough. From the graph, we can see that the maximum value is 8 and the minimum is -4. Therefore, the amplitude is
`\((8 - (-4))/2 = 6\).`

2. Determine the Period: The period of a trigonometric function is the distance along the x-axis to complete one full cycle. On the graph, one full cycle appears to go from 0 to
`\(2\pi\),` which is typical for the sine and cosine functions without any horizontal stretch or compression.

3. Calculate the Phase Shift: The phase shift is the horizontal movement of the function. A cosine function starts at its maximum value. By observing the graph, it seems the first maximum point from the y-axis is at
`\( \pi \),`which indicates a phase shift of
`\( \pi \)` to the right.

4. Determine the Vertical Shift: This is the movement of the function up or down on the graph. The midline of the function appears to be at y = 2 , because that's halfway between the maximum of 8 and the minimum of -4. So there is a vertical shift of 2 units up.

5. Formulate the Function: A general cosine function can be written as:

`\[ f(x) = A\cos(B(x - C)) + D \]`

where A is the amplitude,
`\( (2\pi)/(B) \)`is the period, C is the phase shift, and D is the vertical shift.

Given our observations:

- Amplitude A = 6

- Period
`(\( (2\pi)/(B) \)) = \( 2\pi \)` (since there is no horizontal stretch, B = 1

- Phase Shift C =
`\( \pi \)`

- Vertical Shift D = 2

Plugging these into the general form gives us:

`\[ f(x) = 6\cos(x - \pi) + 2 \]`

This would be the cosine equation of the function shown on the graph. The function has an amplitude of 6, no horizontal stretch (since the period is
`\( 2\pi \)),` a phase shift to the right by
`\( \pi \),` and a vertical shift upwards by 2.