Question

Choose the graph of the following function:

f (x) = cosine (three-halves x)
Graph A
On a coordinate plane, a function curves up through (0, 0) to (0.5, 1), and then curves down through (2, 0) to (3, negative 1). The function crosses the x-axis at (4, 0), (6, 0), and (8, 0).
Graph B
On a coordinate plane, a function curves up through (0, 1) and down through (1, 0) to (2, negative 1). The function continues to curve up through (3, 0) and up to (4, 1).
Graph C
On a coordinate plane, a function has a maximum point at (0, 2) and then curves down through (3, 0) to (6, negative 2) before increasing through the x-axis at (9, 0).
Graph D
On a coordinate plane, a curve increases through (0, 0) to (3, 1) and then decreases through (6, 0).
a.
Graph A
b.
Graph B
c.
Graph C
d.
Graph D

Answer 1

Explanation:

Let's analyze each of the given graphs and match them with the corresponding function description:

a. Graph A:

The function starts at (0, 0), curves up to (0.5, 1), and then curves down through (2, 0) to (3, -1).

The function crosses the x-axis at (4, 0), (6, 0), and (8, 0).

b. Graph B:

The function starts at (0, 1), curves down through (1, 0) to (2, -1).

It continues to curve up through (3, 0) and up to (4, 1).

c. Graph C:

The function has a maximum point at (0, 2), then curves down through (3, 0) to (6, -2).

It increases through the x-axis at (9, 0).

d. Graph D:

The curve increases through (0, 0) to (3, 1) and then decreases through (6, 0).

Now, let's match each graph to its corresponding description:

a. Graph A matches the description.

b. Graph B matches the description.

c. Graph C does not match the description; it has a maximum point, but the behavior does not match the given description.

d. Graph D matches the description.

So, the correct matches are:

a. Graph A

b. Graph B

d. Graph D

Answer 2

Graph B

Explanation:

The graph of f(x) = cos⁡(3x/2) is a result of horizontally compressing the parent function y = cos⁡(x) by a scale factor of 3/2. This compression causes the graph to become narrower and cycle through its oscillation more quickly compared to the parent function.

y-intercept and range

The graph of y = cos(x) has a y-intercept at (0, 1). It oscillates periodically between - 1 and 1 as x varies, so its range is -1 ≤ cos(x) ≤ 1.

The graph of f(x) = cos⁡(3x/2) has undergone a horizontal compression only, so its y-intercept and range are the same as those of the parent function.

x-intercepts of f(x)

To find the x-coordinates of the x-intercepts of the graph of f(x) = cos⁡(3x/2), we set the function to zero and solve for x:

`\cos\left((3x)/(2)\right)=0`

`(3x)/(2)=\cos^(-1)(0)`

`(3x)/(2)=(\pi)/(2)+2\pi n,\;\;(3\pi)/(2)+2\pi n`

`x=(\pi)/(3)+(4\pi n)/(3) ,\;\;\pi+(4\pi n)/(3)`

Therefore, the function crosses the x-axis at the following points (where the x-coordinate is rounded to the nearest integer):

• (1, 0), (3, 0), (5, 0), ...

Minimum points of f(x)

To find the x-coordinates of the minimum points, we set the function to -1 and solve for x:

`\cos\left((3x)/(2)\right)=-1`

`(3x)/(2)=\cos^(-1)(-1)`

`(3x)/(2)=\pi +2\pi n`

`x=(2\pi)/(3)+(4\pi n)/(3)`

Therefore, the function has minimum points at the following points (where the x-coordinate is rounded to the nearest integer):

• (2, -1), (6, -1), (10, -1), ...

Maximum points of f(x)

To find the x-coordinates of the maximum points, we set the function to 1 and solve for x:

`\cos\left((3x)/(2)\right)=1`

`(3x)/(2)=\cos^(-1)(1)`

`(3x)/(2)=2\pi n`

`x=(4\pi n)/(3)`

Therefore, the function's has maximum points at the following points (where the x-coordinate is rounded to the nearest integer):

• (4, 1), (8, 1), (13, 1), ...

Conclusion

Therefore, the graph of function f(x) = cos(3x/2) is:

• Graph B
  On a coordinate plane, a function curves up through (0, 1) and down through (1, 0) to (2, -1). The function continues to curve up through (3, 0) and up to (4, 1).