Question
A graph of a trigonometric function is shown on a coordinate plane. Each branch of the graph begins at positive infinity, passes through the x-axis, and decreases to negative infinity. There are vertical asymptotes at equal intervals.
Question
Use the graph to answer the question.
Which of the following can be the parent function of the graph? Select all that apply.
(1 point)
Responses
sin x
sin x
cos x
cos x
tan x
tan x
cot x
cot x
sec x
sec x
csc x
Question
Use the graph to answer the question.
Which of the following can be the parent function of the graph? Select all that apply.
(1 point)
Responses
sin x
sin x
cos x
cos x
tan x
tan x
cot x
cot x
sec x
sec x
csc x
Answers
GPT-4o mini
To determine which of the given functions can be the parent function of the graph described, we should analyze the properties of each trigonometric function listed:
1. **sin x**: This function oscillates between -1 and 1 and does not have vertical asymptotes. We can eliminate this option.
2. **cos x**: Like sine, cosine oscillates between -1 and 1 and does not have vertical asymptotes. This option is also eliminated.
3. **tan x**: The tangent function has vertical asymptotes at odd multiples of π/2 (i.e., x = (2n+1)π/2 for n ∈ ℤ), begins at negative infinity, passes through the x-axis, and then goes to positive infinity. This matches your description well.
4. **cot x**: The cotangent function has vertical asymptotes at integer multiples of π (i.e., x = nπ for n ∈ ℤ). It also starts at positive infinity, passes through the x-axis, and then decreases to negative infinity. This also fits your description.
5. **sec x**: The secant function has vertical asymptotes at odd multiples of π/2 (like the tangent function) and ranges from -∞ to -1 and 1 to +∞. This function does not match your description since it does not go to +∞ starting at initial points.
6. **csc x**: The cosecant function has vertical asymptotes at integer multiples of π and behaves similarly to secant, but it ranges from -∞ to -1 and 1 to +∞, so it doesn't match the behavior indicated in your description.
**Conclusion**: The potential parent functions based on the description of the graph are:
- tan x
- cot x
These are the functions that can be the parent function of the graph described.
1. **sin x**: This function oscillates between -1 and 1 and does not have vertical asymptotes. We can eliminate this option.
2. **cos x**: Like sine, cosine oscillates between -1 and 1 and does not have vertical asymptotes. This option is also eliminated.
3. **tan x**: The tangent function has vertical asymptotes at odd multiples of π/2 (i.e., x = (2n+1)π/2 for n ∈ ℤ), begins at negative infinity, passes through the x-axis, and then goes to positive infinity. This matches your description well.
4. **cot x**: The cotangent function has vertical asymptotes at integer multiples of π (i.e., x = nπ for n ∈ ℤ). It also starts at positive infinity, passes through the x-axis, and then decreases to negative infinity. This also fits your description.
5. **sec x**: The secant function has vertical asymptotes at odd multiples of π/2 (like the tangent function) and ranges from -∞ to -1 and 1 to +∞. This function does not match your description since it does not go to +∞ starting at initial points.
6. **csc x**: The cosecant function has vertical asymptotes at integer multiples of π and behaves similarly to secant, but it ranges from -∞ to -1 and 1 to +∞, so it doesn't match the behavior indicated in your description.
**Conclusion**: The potential parent functions based on the description of the graph are:
- tan x
- cot x
These are the functions that can be the parent function of the graph described.