Question
A graph of a trigonometric function is shown on a coordinate plane.The graph has asymptotes at x equals negative Start Fraction 3 pi over 2 End Fraction and x equals Start Fraction 3 pi over 2 End Fraction and a midline at y equals 2. Between the asymptotes, the graph increases from negative infinity to the point left-parenthesis 0 comma 1 right-parenthesis and then decreases to negative infinity. From x equals Start Fraction 3 pi over 2 End Fraction to Start Fraction 9 pi over 2. the graph decreases from negative infinity to the point left-parenthesis 3 pi comma 3 right-parenthesis and then increases to infinity. This pattern continues to the left and right.
Question
Use the graph to answer the question.
Which of the following equations can represent the graph? Select all that apply.
(2 points)
Responses
f left parenthesis x right parenthesis equals negative secant Start Fraction x over 3 End Fraction plus 2
Image with alt text: f left parenthesis x right parenthesis equals negative secant Start Fraction x over 3 End Fraction plus 2
f left parenthesis x right parenthesis equals secant Start Fraction x over 3 End Fraction plus 2
Image with alt text: f left parenthesis x right parenthesis equals secant Start Fraction x over 3 End Fraction plus 2
f left parenthesis x right parenthesis equals negative secant left parenthesis Start Fraction x over 3 End Fraction plus pi right parenthesis plus 2
Image with alt text: f left parenthesis x right parenthesis equals negative secant left parenthesis Start Fraction x over 3 End Fraction plus pi right parenthesis plus 2
f left parenthesis x right parenthesis equals secant left parenthesis Start Fraction x over 3 End Fraction plus pi right parenthesis plus 2
Image with alt text: f left parenthesis x right parenthesis equals secant left parenthesis Start Fraction x over 3 End Fraction plus pi right parenthesis plus 2
f left parenthesis x right parenthesis equals negative cosecant left parenthesis Start Fraction x over 3 End Fraction plus Start Fraction 3 pi over 2 End Fraction right parenthesis plus 2
Image with alt text: f left parenthesis x right parenthesis equals negative cosecant left parenthesis Start Fraction x over 3 End Fraction plus Start Fraction 3 pi over 2 End Fraction right parenthesis plus 2
f left parenthesis x right parenthesis equals cosecant left parenthesis Start Fraction x over 3 End Fraction plus Start Fraction 3 pi over 2 End Fraction right parenthesis plus 2
Image with alt text: f left parenthesis x right parenthesis equals cosecant left parenthesis Start Fraction x over 3 End Fraction plus Start Fraction 3 pi over 2 End Fraction right parenthesis plus 2
Question
Use the graph to answer the question.
Which of the following equations can represent the graph? Select all that apply.
(2 points)
Responses
f left parenthesis x right parenthesis equals negative secant Start Fraction x over 3 End Fraction plus 2
Image with alt text: f left parenthesis x right parenthesis equals negative secant Start Fraction x over 3 End Fraction plus 2
f left parenthesis x right parenthesis equals secant Start Fraction x over 3 End Fraction plus 2
Image with alt text: f left parenthesis x right parenthesis equals secant Start Fraction x over 3 End Fraction plus 2
f left parenthesis x right parenthesis equals negative secant left parenthesis Start Fraction x over 3 End Fraction plus pi right parenthesis plus 2
Image with alt text: f left parenthesis x right parenthesis equals negative secant left parenthesis Start Fraction x over 3 End Fraction plus pi right parenthesis plus 2
f left parenthesis x right parenthesis equals secant left parenthesis Start Fraction x over 3 End Fraction plus pi right parenthesis plus 2
Image with alt text: f left parenthesis x right parenthesis equals secant left parenthesis Start Fraction x over 3 End Fraction plus pi right parenthesis plus 2
f left parenthesis x right parenthesis equals negative cosecant left parenthesis Start Fraction x over 3 End Fraction plus Start Fraction 3 pi over 2 End Fraction right parenthesis plus 2
Image with alt text: f left parenthesis x right parenthesis equals negative cosecant left parenthesis Start Fraction x over 3 End Fraction plus Start Fraction 3 pi over 2 End Fraction right parenthesis plus 2
f left parenthesis x right parenthesis equals cosecant left parenthesis Start Fraction x over 3 End Fraction plus Start Fraction 3 pi over 2 End Fraction right parenthesis plus 2
Image with alt text: f left parenthesis x right parenthesis equals cosecant left parenthesis Start Fraction x over 3 End Fraction plus Start Fraction 3 pi over 2 End Fraction right parenthesis plus 2
Answers
GPT-4o mini
To determine which equations can represent the graph described, we need to analyze the properties outlined:
1. **Asymptotes**: The vertical asymptotes at \( x = -\frac{3\pi}{2} \) and \( x = \frac{3\pi}{2} \) suggest that the function could be a secant or cosecant function, as these functions have vertical asymptotes.
2. **Midline**: The midline is at \( y = 2 \), indicating that the function oscillates around this line. Therefore, any function that represents this graph must have a vertical shift of +2.
3. **Behavior between asymptotes**: The graph has the following pattern:
- From \( x = -\frac{3\pi}{2} \) to \( x = 0 \): the graph increases from negative infinity to the point \( (0, 1) \) and then decreases to negative infinity.
- From \( x = \frac{3\pi}{2} \) to \( x = 3\pi \): the graph decreases from negative infinity to the point \( (3\pi, 3) \) and then increases to positive infinity.
4. **Function Type**: The described behavior around the asymptotes and midline is characteristic of a negative secant function, since it opens downward between its asymptotes.
### Given Options:
- **(1)** \( f(x) = -\sec\left(\frac{x}{3}\right) + 2 \)
- **(2)** \( f(x) = \sec\left(\frac{x}{3}\right) + 2 \)
- **(3)** \( f(x) = -\sec\left(\frac{x}{3} + \pi\right) + 2 \)
- **(4)** \( f(x) = \sec\left(\frac{x}{3} + \pi\right) + 2 \)
- **(5)** \( f(x) = -\csc\left(\frac{x}{3} + \frac{3\pi}{2}\right) + 2 \)
- **(6)** \( f(x) = \csc\left(\frac{x}{3} + \frac{3\pi}{2}\right) + 2 \)
### Analysis of Options:
- **(1)** and **(3)**: Both involve negative secant functions which fit the behavior of the graph, as both of these have a midline shift of +2. The periodic nature of secant functions aligns with the observed pattern of increasing and decreasing.
- **(2)** and **(4)**: Both options involve positive secant functions, which do not align with the description of the graph that decreases to negative infinity after reaching a local maximum.
- **(5)** and **(6)**: Both involve cosecant functions, but the description of the behavior indicates that we are dealing with secant functions rather than cosecant functions.
### Conclusion:
The equations that can represent the graph are:
- **\( f(x) = -\sec\left(\frac{x}{3}\right) + 2 \)**
- **\( f(x) = -\sec\left(\frac{x}{3} + \pi\right) + 2 \)**
Thus, the valid options are **1 and 3**.
1. **Asymptotes**: The vertical asymptotes at \( x = -\frac{3\pi}{2} \) and \( x = \frac{3\pi}{2} \) suggest that the function could be a secant or cosecant function, as these functions have vertical asymptotes.
2. **Midline**: The midline is at \( y = 2 \), indicating that the function oscillates around this line. Therefore, any function that represents this graph must have a vertical shift of +2.
3. **Behavior between asymptotes**: The graph has the following pattern:
- From \( x = -\frac{3\pi}{2} \) to \( x = 0 \): the graph increases from negative infinity to the point \( (0, 1) \) and then decreases to negative infinity.
- From \( x = \frac{3\pi}{2} \) to \( x = 3\pi \): the graph decreases from negative infinity to the point \( (3\pi, 3) \) and then increases to positive infinity.
4. **Function Type**: The described behavior around the asymptotes and midline is characteristic of a negative secant function, since it opens downward between its asymptotes.
### Given Options:
- **(1)** \( f(x) = -\sec\left(\frac{x}{3}\right) + 2 \)
- **(2)** \( f(x) = \sec\left(\frac{x}{3}\right) + 2 \)
- **(3)** \( f(x) = -\sec\left(\frac{x}{3} + \pi\right) + 2 \)
- **(4)** \( f(x) = \sec\left(\frac{x}{3} + \pi\right) + 2 \)
- **(5)** \( f(x) = -\csc\left(\frac{x}{3} + \frac{3\pi}{2}\right) + 2 \)
- **(6)** \( f(x) = \csc\left(\frac{x}{3} + \frac{3\pi}{2}\right) + 2 \)
### Analysis of Options:
- **(1)** and **(3)**: Both involve negative secant functions which fit the behavior of the graph, as both of these have a midline shift of +2. The periodic nature of secant functions aligns with the observed pattern of increasing and decreasing.
- **(2)** and **(4)**: Both options involve positive secant functions, which do not align with the description of the graph that decreases to negative infinity after reaching a local maximum.
- **(5)** and **(6)**: Both involve cosecant functions, but the description of the behavior indicates that we are dealing with secant functions rather than cosecant functions.
### Conclusion:
The equations that can represent the graph are:
- **\( f(x) = -\sec\left(\frac{x}{3}\right) + 2 \)**
- **\( f(x) = -\sec\left(\frac{x}{3} + \pi\right) + 2 \)**
Thus, the valid options are **1 and 3**.