Question

Compare the functions shown below:

g(x)
f(x) = −6x − 3 cosine function with y intercept at 0, negative 3 h(x) = 2 cos(x + π) − 1


Using complete sentences, explain which function has the greatest y-intercept.

Answer 1

Both functions have the same y-intercepts.

• y = -3

Answer 2

All functions have the same y-intercept of
`\( -3 \)`, and none has a greater y-intercept than the others based on the information provided.

To determine which function has the greatest y-intercept, we need to evaluate the y-intercept for each given function. The y-intercept of a function is the value of the function when
`\( x = 0 \)`.

For
`\( f(x) = -6x - 3 \)`:

The y-intercept can be found by setting
`\( x = 0 \)`:

`\( f(0) = -6(0) - 3 = -3 \)`

So, the y-intercept for
`\( f(x) \)` is
`\( -3 \)`.

For
`\( h(x) = 2 \cos(x + \pi) - 1 \)`:

The y-intercept can be found by setting
`\( x = 0 \)`:

`\( h(0) = 2 \cos(0 + \pi) - 1 = 2 \cos(\pi) - 1 = 2(-1) - 1 = -2 - 1 = -3 \)`

So, the y-intercept for
`\( h(x) \)` is
`\( -3 \)\( -3 \)`.

Since both functions have the same y-intercept of
`\( -3 \)`, neither function has a greater y-intercept than the other; they are equal.

The function \( g(x) \) is not given explicitly, but if we assume it is the cosine function graphed in the image with a y-intercept at
`\( (0, -3) \)`, then it would also have a y-intercept of
`\( -3 \)`, the same as the other two functions.

Therefore, all functions have the same y-intercept of
`\( -3 \)`, and none has a greater y-intercept than the others based on the information provided.