Question
The graph of a trigonometric function is shown. The graph oscillates between y equals 2.5 and y equals negative 3.5. Its midline is located at y equals negative one-half. The graph crosses the midline at x values 1 less than integer multiples of pi.
Question
Use the graph to answer the question.
The graph is supposed to show f left parenthesis x right parenthesis equals 3 sine left parenthesis Start Fraction x over 4 End Fraction plus 1 right parenthesis minus one half. Which of the following are correctly represented in the graph? Select all that apply.
(1 point)
Responses
the amplitude
the amplitude
the vertical shift
the vertical shift
the horizontal shift
the horizontal shift
the period
the period
the horizontal expansion or compression
Question
Use the graph to answer the question.
The graph is supposed to show f left parenthesis x right parenthesis equals 3 sine left parenthesis Start Fraction x over 4 End Fraction plus 1 right parenthesis minus one half. Which of the following are correctly represented in the graph? Select all that apply.
(1 point)
Responses
the amplitude
the amplitude
the vertical shift
the vertical shift
the horizontal shift
the horizontal shift
the period
the period
the horizontal expansion or compression
Answers
GPT-4o mini
To analyze the trigonometric function \( f(x) = 3 \sin\left(\frac{x}{4} + 1\right) - \frac{1}{2} \), let's break it down:
1. **Amplitude**: The amplitude of a sine function \( A \sin(B(x - C)) + D \) is given by the absolute value of \( A \). In this case, the amplitude is \( |3| = 3 \).
2. **Vertical Shift**: The vertical shift is determined by \( D \). Here, we have \( D = -\frac{1}{2} \), indicating that the entire graph is shifted downwards by \( \frac{1}{2} \) units.
3. **Horizontal Shift**: The sine function can be horizontally shifted by \( C \). In this case, the term \( +1 \) inside the sine function is a shift to the left. However, to express this in terms of \( C \), you would typically factor it out as \( \sin\left(\frac{x}{4} + 1\right) = \sin\left(\frac{x}{4} + \frac{4}{4}\right) \) which represents a leftward shift but in a more complicated way to interpret directly. This does not directly imply a clean horizontal shift like it would with \( (x - C) \).
4. **Period**: The period of the sine function is given by \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) in the sine function. Here, since \( B = \frac{1}{4} \), the period is \( \frac{2\pi}{1/4} = 8\pi \).
5. **Horizontal Expansion or Compression**: The factor \( \frac{1}{4} \) in front of \( x \) indicates a horizontal expansion, meaning the graph stretches along the x-axis, resulting in a longer period.
Based on the analysis above, the following features are correctly represented in the graph:
- The amplitude: **Yes**
- The vertical shift: **Yes**
- The horizontal shift: **No** (the way it’s expressed does not clearly define a shift)
- The period: **Yes**
- The horizontal expansion or compression: **Yes**
**Correct selections**: Amplitude, Vertical Shift, Period, Horizontal Expansion or Compression.
1. **Amplitude**: The amplitude of a sine function \( A \sin(B(x - C)) + D \) is given by the absolute value of \( A \). In this case, the amplitude is \( |3| = 3 \).
2. **Vertical Shift**: The vertical shift is determined by \( D \). Here, we have \( D = -\frac{1}{2} \), indicating that the entire graph is shifted downwards by \( \frac{1}{2} \) units.
3. **Horizontal Shift**: The sine function can be horizontally shifted by \( C \). In this case, the term \( +1 \) inside the sine function is a shift to the left. However, to express this in terms of \( C \), you would typically factor it out as \( \sin\left(\frac{x}{4} + 1\right) = \sin\left(\frac{x}{4} + \frac{4}{4}\right) \) which represents a leftward shift but in a more complicated way to interpret directly. This does not directly imply a clean horizontal shift like it would with \( (x - C) \).
4. **Period**: The period of the sine function is given by \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( x \) in the sine function. Here, since \( B = \frac{1}{4} \), the period is \( \frac{2\pi}{1/4} = 8\pi \).
5. **Horizontal Expansion or Compression**: The factor \( \frac{1}{4} \) in front of \( x \) indicates a horizontal expansion, meaning the graph stretches along the x-axis, resulting in a longer period.
Based on the analysis above, the following features are correctly represented in the graph:
- The amplitude: **Yes**
- The vertical shift: **Yes**
- The horizontal shift: **No** (the way it’s expressed does not clearly define a shift)
- The period: **Yes**
- The horizontal expansion or compression: **Yes**
**Correct selections**: Amplitude, Vertical Shift, Period, Horizontal Expansion or Compression.