Question
Graph the following points on the graphing calculator:
(0, 6), (StartFraction pi over 2, 7), (pi, 8), (StartFraction 3pi over 2, 7), and (2pi, 6)
Explain how to use the graph to write an equation to model the gum’s height. Be sure to identify the pattern of the points in your explanation, and identify the values of a and k.
(0, 6), (StartFraction pi over 2, 7), (pi, 8), (StartFraction 3pi over 2, 7), and (2pi, 6)
Explain how to use the graph to write an equation to model the gum’s height. Be sure to identify the pattern of the points in your explanation, and identify the values of a and k.
Answers
GPT-4o mini
To model the height of the gum based on the points provided, we first note the coordinates:
- (0, 6)
- \(\left(\frac{\pi}{2}, 7\right)\)
- (\(\pi, 8\))
- \(\left(\frac{3\pi}{2}, 7\right)\)
- (2\(\pi\), 6)
### Observing the Pattern
1. **Variation in Height**:
- The heights at these points are changing: starting from 6, peaking at 8, and returning back to 6.
- The maximum height occurs at \(\pi\) where the height is 8, and the heights at \(0\), \(2\pi\) return to 6.
2. **Symmetry**:
- The graph is symmetric around the point \((\pi, 8)\) which suggests a periodic nature.
3. **Periodicity**:
- The values of \(y\) appear to cycle every \(2\pi\), indicating that the function likely has a period of \(2\pi\).
### Function Model
Given that the height of the gum follows a sinusoidal pattern, we can use a cosine function to model the height:
\[
h(t) = a \cos(k(t - d)) + b
\]
Where:
- \(a\) is the amplitude,
- \(k\) affects the period,
- \(d\) is the horizontal shift,
- \(b\) is the vertical shift.
### Finding Parameters
1. **Amplitude (\(a\))**:
- The maximum height is 8 and the minimum height is 6. Therefore, the amplitude is:
\[
a = \frac{\text{max} - \text{min}}{2} = \frac{8 - 6}{2} = 1
\]
2. **Vertical Shift (\(b\))**:
- The average of the maximum and minimum values gives the vertical shift:
\[
b = \frac{\text{max} + \text{min}}{2} = \frac{8 + 6}{2} = 7
\]
3. **Period (\(k\))**:
- The period \(T\) of the function can be found from the points provided. The points are one full cycle apart (from \(0\) to \(2\pi\)):
\[
T = 2\pi
\]
- The formula for the period in terms of \(k\) is \(T = \frac{2\pi}{k}\). Hence,
\[
k = 1
\]
### Horizontal Shift (\(d\))
In this case, the cosine function starts at its maximum. Therefore, we do not need a horizontal shift:
- \(d = 0\).
### Final Equation
Putting it all together, the equation that models the height of the gum based on these points is:
\[
h(t) = 1 \cos(1(t)) + 7
\]
Or simplified:
\[
h(t) = \cos(t) + 7
\]
This equation correctly reflects the periodic behavior and maximum and minimum heights observed in the points given.
- (0, 6)
- \(\left(\frac{\pi}{2}, 7\right)\)
- (\(\pi, 8\))
- \(\left(\frac{3\pi}{2}, 7\right)\)
- (2\(\pi\), 6)
### Observing the Pattern
1. **Variation in Height**:
- The heights at these points are changing: starting from 6, peaking at 8, and returning back to 6.
- The maximum height occurs at \(\pi\) where the height is 8, and the heights at \(0\), \(2\pi\) return to 6.
2. **Symmetry**:
- The graph is symmetric around the point \((\pi, 8)\) which suggests a periodic nature.
3. **Periodicity**:
- The values of \(y\) appear to cycle every \(2\pi\), indicating that the function likely has a period of \(2\pi\).
### Function Model
Given that the height of the gum follows a sinusoidal pattern, we can use a cosine function to model the height:
\[
h(t) = a \cos(k(t - d)) + b
\]
Where:
- \(a\) is the amplitude,
- \(k\) affects the period,
- \(d\) is the horizontal shift,
- \(b\) is the vertical shift.
### Finding Parameters
1. **Amplitude (\(a\))**:
- The maximum height is 8 and the minimum height is 6. Therefore, the amplitude is:
\[
a = \frac{\text{max} - \text{min}}{2} = \frac{8 - 6}{2} = 1
\]
2. **Vertical Shift (\(b\))**:
- The average of the maximum and minimum values gives the vertical shift:
\[
b = \frac{\text{max} + \text{min}}{2} = \frac{8 + 6}{2} = 7
\]
3. **Period (\(k\))**:
- The period \(T\) of the function can be found from the points provided. The points are one full cycle apart (from \(0\) to \(2\pi\)):
\[
T = 2\pi
\]
- The formula for the period in terms of \(k\) is \(T = \frac{2\pi}{k}\). Hence,
\[
k = 1
\]
### Horizontal Shift (\(d\))
In this case, the cosine function starts at its maximum. Therefore, we do not need a horizontal shift:
- \(d = 0\).
### Final Equation
Putting it all together, the equation that models the height of the gum based on these points is:
\[
h(t) = 1 \cos(1(t)) + 7
\]
Or simplified:
\[
h(t) = \cos(t) + 7
\]
This equation correctly reflects the periodic behavior and maximum and minimum heights observed in the points given.