Question
A scatter plot is shown with time in seconds on the x-axis and height in feet on the y-axis.Points are plotted at left-parenthesis 0 comma 3 right-parenthesis, left-parenthesis 1 comma 1.25 right-parenthesis, left-parenthesis 2 comma 0.5 right-parenthesis, left-parenthesis 3 comma 0 right-parenthesis, left-parenthesis 4 comma 0.25 right-parenthesis, left-parenthesis 5 comma 1.75 right-parenthesis, left-parenthesis 6 comma 3.25 right-parenthesis, left-parenthesis 7 comma 4.25 right-parenthesis, left-parenthesis 8 comma 5.75 right-parenthesis, left-parenthesis 9 comma 6 right-parenthesis, left-parenthesis 10 comma 5.25 right-parenthesis, left-parenthesis 11 comma 4 right-parenthesis, left-parenthesis 12 comma 3.25 right-parenthesis. The points decrease to left-parenthesis 15 comma 0.1 right-parenthesis then increase to left-parenthesis 21 comma 5.8 right-parenthesis and then decrease to left-parenthesis 23 comma 4.
Question
Use the graph to answer the question.
The height of the tide over time is shown on the graph. Which of the following equations can be used to reasonably model the data represented in the graph? Select all that apply.
(1 point)
Responses
f left parenthesis x right parenthesis equals negative 3 sine left parenthesis Start Fraction pi over 6 End Fraction x right parenthesis plus 3
Image with alt text: f left parenthesis x right parenthesis equals negative 3 sine left parenthesis Start Fraction pi over 6 End Fraction x right parenthesis plus 3
f left parenthesis x right parenthesis equals 3 sine Start Fraction pi over 6 End Fraction x plus 3
Image with alt text: f left parenthesis x right parenthesis equals 3 sine Start Fraction pi over 6 End Fraction x plus 3
f left parenthesis x right parenthesis equals negative 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x right parenthesis plus 3
Image with alt text: f left parenthesis x right parenthesis equals negative 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x right parenthesis plus 3
f left parenthesis x right parenthesis equals 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x plus Start Fraction pi over 2 End Fraction right parenthesis plus 3
Image with alt text: f left parenthesis x right parenthesis equals 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x plus Start Fraction pi over 2 End Fraction right parenthesis plus 3
f left parenthesis x right parenthesis equals negative 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x plus Start Fraction pi over 2 End Fraction right parenthesis plus 3
Image with alt text: f left parenthesis x right parenthesis equals negative 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x plus Start Fraction pi over 2 End Fraction right parenthesis plus 3
Question
Use the graph to answer the question.
The height of the tide over time is shown on the graph. Which of the following equations can be used to reasonably model the data represented in the graph? Select all that apply.
(1 point)
Responses
f left parenthesis x right parenthesis equals negative 3 sine left parenthesis Start Fraction pi over 6 End Fraction x right parenthesis plus 3
Image with alt text: f left parenthesis x right parenthesis equals negative 3 sine left parenthesis Start Fraction pi over 6 End Fraction x right parenthesis plus 3
f left parenthesis x right parenthesis equals 3 sine Start Fraction pi over 6 End Fraction x plus 3
Image with alt text: f left parenthesis x right parenthesis equals 3 sine Start Fraction pi over 6 End Fraction x plus 3
f left parenthesis x right parenthesis equals negative 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x right parenthesis plus 3
Image with alt text: f left parenthesis x right parenthesis equals negative 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x right parenthesis plus 3
f left parenthesis x right parenthesis equals 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x plus Start Fraction pi over 2 End Fraction right parenthesis plus 3
Image with alt text: f left parenthesis x right parenthesis equals 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x plus Start Fraction pi over 2 End Fraction right parenthesis plus 3
f left parenthesis x right parenthesis equals negative 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x plus Start Fraction pi over 2 End Fraction right parenthesis plus 3
Image with alt text: f left parenthesis x right parenthesis equals negative 3 cosine left parenthesis Start Fraction pi over 6 End Fraction x plus Start Fraction pi over 2 End Fraction right parenthesis plus 3
Answers
Answered by
GPT-4o mini
To determine which equations can reasonably model the tide height data over time, we should analyze the properties of the sinusoidal functions represented by the equations provided.
1. **Understanding the Graph**:
- The height data shows periodic behavior, typical of tidal patterns, which suggests a sine or cosine function could be an appropriate model.
- The tide appears to rise and fall; the values suggest a vertical shift and amplitude.
2. **Evaluating Each Function**:
- **(1)** \( f(x) = -3 \sin\left(\frac{\pi}{6} x\right) + 3 \)
- This function starts at its maximum (because of the negative sine) at \( x = 0 \) with a vertical shift of +3, which could match the given height.
- **(2)** \( f(x) = 3 \sin\left(\frac{\pi}{6} x\right) + 3 \)
- This function starts at 3 when \( x = 0\) but rises first, not matching the observed start of the curve.
- **(3)** \( f(x) = -3 \cos\left(\frac{\pi}{6} x\right) + 3 \)
- This looks like it starts at its maximum due to the negative cosine, which could fit since the starting height is 3.
- **(4)** \( f(x) = 3 \cos\left(\frac{\pi}{6} x + \frac{\pi}{2}\right) + 3 \)
- The term \( \frac{\pi}{2} \) within the cosine shifts it to start at 0, which does not correctly match with observed data.
- **(5)** \( f(x) = -3 \cos\left(\frac{\pi}{6} x + \frac{\pi}{2}\right) + 3 \)
- Similar to (4), it starts at 3 due to the cosine shifted by \( \frac{\pi}{2} \), so it reduces quickly, matching the decreasing height initially.
3. **Selection of Correct Models**:
- The plausible models that fit the observed behavior around the starting height and subsequent behavior:
- **(1)** \( f(x) = -3 \sin\left(\frac{\pi}{6} x\right) + 3 \)
- **(3)** \( f(x) = -3 \cos\left(\frac{\pi}{6} x\right) + 3 \)
- **(5)** \( f(x) = -3 \cos\left(\frac{\pi}{6} x + \frac{\pi}{2}\right) + 3 \)
### Final Answers:
The equations that can reasonably model the data represented in the graph are:
- \( f(x) = -3 \sin\left(\frac{\pi}{6} x\right) + 3 \) (1)
- \( f(x) = -3 \cos\left(\frac{\pi}{6} x\right) + 3 \) (3)
- \( f(x) = -3 \cos\left(\frac{\pi}{6} x + \frac{\pi}{2}\right) + 3 \) (5)
1. **Understanding the Graph**:
- The height data shows periodic behavior, typical of tidal patterns, which suggests a sine or cosine function could be an appropriate model.
- The tide appears to rise and fall; the values suggest a vertical shift and amplitude.
2. **Evaluating Each Function**:
- **(1)** \( f(x) = -3 \sin\left(\frac{\pi}{6} x\right) + 3 \)
- This function starts at its maximum (because of the negative sine) at \( x = 0 \) with a vertical shift of +3, which could match the given height.
- **(2)** \( f(x) = 3 \sin\left(\frac{\pi}{6} x\right) + 3 \)
- This function starts at 3 when \( x = 0\) but rises first, not matching the observed start of the curve.
- **(3)** \( f(x) = -3 \cos\left(\frac{\pi}{6} x\right) + 3 \)
- This looks like it starts at its maximum due to the negative cosine, which could fit since the starting height is 3.
- **(4)** \( f(x) = 3 \cos\left(\frac{\pi}{6} x + \frac{\pi}{2}\right) + 3 \)
- The term \( \frac{\pi}{2} \) within the cosine shifts it to start at 0, which does not correctly match with observed data.
- **(5)** \( f(x) = -3 \cos\left(\frac{\pi}{6} x + \frac{\pi}{2}\right) + 3 \)
- Similar to (4), it starts at 3 due to the cosine shifted by \( \frac{\pi}{2} \), so it reduces quickly, matching the decreasing height initially.
3. **Selection of Correct Models**:
- The plausible models that fit the observed behavior around the starting height and subsequent behavior:
- **(1)** \( f(x) = -3 \sin\left(\frac{\pi}{6} x\right) + 3 \)
- **(3)** \( f(x) = -3 \cos\left(\frac{\pi}{6} x\right) + 3 \)
- **(5)** \( f(x) = -3 \cos\left(\frac{\pi}{6} x + \frac{\pi}{2}\right) + 3 \)
### Final Answers:
The equations that can reasonably model the data represented in the graph are:
- \( f(x) = -3 \sin\left(\frac{\pi}{6} x\right) + 3 \) (1)
- \( f(x) = -3 \cos\left(\frac{\pi}{6} x\right) + 3 \) (3)
- \( f(x) = -3 \cos\left(\frac{\pi}{6} x + \frac{\pi}{2}\right) + 3 \) (5)
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