Question
The tide of the ocean at the shore can be modeled by the following graph.
4-3-2-10
-2
-5
-6
1 2 3 4 5 6 7
8 9
10 11
12 13 14 15 16 17 18
Which of the equations below best represents this graph?
(5 points)
A) y=-3 cos(pie/4 x)
B) y = 3 cos(pie/8 x)
C) y=-3 sin(pie/8 x)
D) y=-3 cos(pie/8 x)
4-3-2-10
-2
-5
-6
1 2 3 4 5 6 7
8 9
10 11
12 13 14 15 16 17 18
Which of the equations below best represents this graph?
(5 points)
A) y=-3 cos(pie/4 x)
B) y = 3 cos(pie/8 x)
C) y=-3 sin(pie/8 x)
D) y=-3 cos(pie/8 x)
Answers
GPT-4o mini
To determine which equation best represents the tide graph you've described, we should identify some key characteristics from the graph:
1. **Amplitude**: The range of the tide values suggests that the amplitude (the height of the wave from the centerline) is around 3. We can see the values oscillating between about 3 and -3, suggesting an amplitude of 3.
2. **Period**: The period of the tide typically describes how long it takes for one complete cycle of tides to happen. If there are 8 complete units of x (horizontal axis) corresponding to the one cycle of tide going from maximum to maximum, we can calculate the period as follows. If visible it appears to complete one full oscillation, that would typically mean something like a cosine or sine wave.
3. **Vertical Shift**: It seems there is no vertical shift since the oscillation is centered around the x-axis.
4. **Starting Point**: Cosine functions typically start at their maximum, while sine functions start at the midline going up.
Given these characteristics:
- The amplitude is indeed 3.
- The period for the cosine function is given by the formula: \( T = \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). If the period is roughly 8 units, we can deduce that \( \frac{2\pi}{b} = 8 \) suggests that \( b = \frac{\pi}{4} \).
Thus, the function would be in the format of:
\[ y = -3 \cos\left(\frac{\pi}{4} x\right) \]
From the options:
A) \( y = -3 \cos\left(\frac{\pi}{4} x\right) \) [matches amplitude, period, and phase]
B) \( y = 3 \cos\left(\frac{\pi}{8} x\right) \) [incorrect amplitude and period]
C) \( y = -3 \sin\left(\frac{\pi}{8} x\right) \) [incorrect function type and period]
D) \( y = -3 \cos\left(\frac{\pi}{8} x\right) \) [incorrect period]
Thus, the option that best represents the tide graph is:
**A) y = -3 cos(π/4 x)**.
1. **Amplitude**: The range of the tide values suggests that the amplitude (the height of the wave from the centerline) is around 3. We can see the values oscillating between about 3 and -3, suggesting an amplitude of 3.
2. **Period**: The period of the tide typically describes how long it takes for one complete cycle of tides to happen. If there are 8 complete units of x (horizontal axis) corresponding to the one cycle of tide going from maximum to maximum, we can calculate the period as follows. If visible it appears to complete one full oscillation, that would typically mean something like a cosine or sine wave.
3. **Vertical Shift**: It seems there is no vertical shift since the oscillation is centered around the x-axis.
4. **Starting Point**: Cosine functions typically start at their maximum, while sine functions start at the midline going up.
Given these characteristics:
- The amplitude is indeed 3.
- The period for the cosine function is given by the formula: \( T = \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). If the period is roughly 8 units, we can deduce that \( \frac{2\pi}{b} = 8 \) suggests that \( b = \frac{\pi}{4} \).
Thus, the function would be in the format of:
\[ y = -3 \cos\left(\frac{\pi}{4} x\right) \]
From the options:
A) \( y = -3 \cos\left(\frac{\pi}{4} x\right) \) [matches amplitude, period, and phase]
B) \( y = 3 \cos\left(\frac{\pi}{8} x\right) \) [incorrect amplitude and period]
C) \( y = -3 \sin\left(\frac{\pi}{8} x\right) \) [incorrect function type and period]
D) \( y = -3 \cos\left(\frac{\pi}{8} x\right) \) [incorrect period]
Thus, the option that best represents the tide graph is:
**A) y = -3 cos(π/4 x)**.