Question
A graph of a trigonometric function is shown on a coordinate plane. The curve intersects the x-axis at negative 3 pi, decreases to y equals negative 1, begins increasing, passes through the origin to y equals 1, begins decreasing, and passes through the x-axis at 3 pi. This pattern continues to the left and right.
Question
Use the graph to answer the question.
Jesse wants to write a function to represent the graph.
He says that the period of the graph is triple that of sin x and that the equation of the graph is f left parenthesis x right parenthesis equals 3 sine x. Why is he incorrect?
(1 point)
Responses
The period of the graph is twice that of sin x.
The period of the graph is twice that of sin x .
The frequency of the graph is triple that of sin x.
The frequency of the graph is triple that of sin x .
The equation of the graph should be f left parenthesis x right parenthesis equals sine StartFraction x over 3 EndFraction.
The equation of the graph should be Image with alt text: f left parenthesis x right parenthesis equals sine StartFraction x over 3 EndFraction .
The equation of the graph should be f left parenthesis x right parenthesis equals 3 sine x.
The equation of the graph should be Image with alt text: f left parenthesis x right parenthesis equals 3 sine x .
Question
Use the graph to answer the question.
Jesse wants to write a function to represent the graph.
He says that the period of the graph is triple that of sin x and that the equation of the graph is f left parenthesis x right parenthesis equals 3 sine x. Why is he incorrect?
(1 point)
Responses
The period of the graph is twice that of sin x.
The period of the graph is twice that of sin x .
The frequency of the graph is triple that of sin x.
The frequency of the graph is triple that of sin x .
The equation of the graph should be f left parenthesis x right parenthesis equals sine StartFraction x over 3 EndFraction.
The equation of the graph should be Image with alt text: f left parenthesis x right parenthesis equals sine StartFraction x over 3 EndFraction .
The equation of the graph should be f left parenthesis x right parenthesis equals 3 sine x.
The equation of the graph should be Image with alt text: f left parenthesis x right parenthesis equals 3 sine x .
Answers
GPT-4o mini
To understand why Jesse is incorrect, we need to analyze the characteristics of the function he proposes and compare them with the properties of the sine function.
1. **Identifying the Period**:
- The standard sine function, \( \sin(x) \), has a period of \( 2\pi \). If Jesse says the period of the graph is triple that of \( \sin(x) \), it would be \( 3 \times 2\pi = 6\pi \).
- However, since the curve intersects the x-axis at intervals of \( -3\pi \) and \( 3\pi \), it suggests a shorter period, likely \( 6\pi \) would not match the observed behavior unless the function is meant to reflect a larger period or an error in period measurement.
2. **Understanding Frequency**:
- Frequency is the reciprocal of the period. If the period were indeed \( 6\pi \), the frequency would be \( \frac{1}{6\pi} \), which is not applicable here.
3. **Function Form**:
- Jesse suggests \( f(x) = 3 \sin(x) \). This suggests an amplitude of 3, meaning the function would oscillate between -3 and 3. However, the graph indicates a maximum value of 1 and a minimum value of -1, contrasting the proposed amplitude.
Thus, the correct response regarding Jesse's misunderstanding is:
**The equation of the graph should be \( f(x) = \sin\left(\frac{x}{3}\right) \).**
This form indicates a period that can match the characteristics of the graph, specifically adjusting input \( x \) by a factor of \( 3 \) to produce the required period change from \( 2\pi \) to \( 6\pi \). Additionally, the improper amplitude stated in \( 3\sin(x) \) does not align with the graph as described.
1. **Identifying the Period**:
- The standard sine function, \( \sin(x) \), has a period of \( 2\pi \). If Jesse says the period of the graph is triple that of \( \sin(x) \), it would be \( 3 \times 2\pi = 6\pi \).
- However, since the curve intersects the x-axis at intervals of \( -3\pi \) and \( 3\pi \), it suggests a shorter period, likely \( 6\pi \) would not match the observed behavior unless the function is meant to reflect a larger period or an error in period measurement.
2. **Understanding Frequency**:
- Frequency is the reciprocal of the period. If the period were indeed \( 6\pi \), the frequency would be \( \frac{1}{6\pi} \), which is not applicable here.
3. **Function Form**:
- Jesse suggests \( f(x) = 3 \sin(x) \). This suggests an amplitude of 3, meaning the function would oscillate between -3 and 3. However, the graph indicates a maximum value of 1 and a minimum value of -1, contrasting the proposed amplitude.
Thus, the correct response regarding Jesse's misunderstanding is:
**The equation of the graph should be \( f(x) = \sin\left(\frac{x}{3}\right) \).**
This form indicates a period that can match the characteristics of the graph, specifically adjusting input \( x \) by a factor of \( 3 \) to produce the required period change from \( 2\pi \) to \( 6\pi \). Additionally, the improper amplitude stated in \( 3\sin(x) \) does not align with the graph as described.