Answer :
To graph the given points and find an equation to model the points, follow these steps:
### Step 1: Plotting the Points
First, let's plot the given points on a coordinate plane. The points are:
- [tex]\( (0, 6) \)[/tex]
- [tex]\( \left(\frac{\pi}{2}, 7\right) \)[/tex]
- [tex]\( (\pi, 8) \)[/tex]
- [tex]\( \left(\frac{3\pi}{2}, 7\right) \)[/tex]
- [tex]\( (2\pi, 6) \)[/tex]
### Step 2: Observing the Pattern
Examining the plotted points, we notice the following pattern:
1. The function has a periodic nature with points repeating approximately every [tex]\(2\pi\)[/tex].
2. The values oscillate around a central value, peaking, and then dipping in a consistent pattern.
### Step 3: Identifying Parameters
Next, identify the parameters of a sinusoidal function. We consider a function of the form:
[tex]\[ y = a \sin(bx + c) + d \][/tex]
From the given points, we can make the following observations:
1. Periodicity: The function completes one cycle over [tex]\(2\pi\)[/tex], suggesting the period [tex]\(T = 2\pi\)[/tex]. The frequency [tex]\(b\)[/tex] then is
[tex]\[ b = \frac{2\pi}{T} = \frac{2\pi}{2\pi} = 1. \][/tex]
2. Amplitude: The height of the peaks and troughs about the midline of the function can be observed. From the points, the maximum value is 8, and the minimum value is 6:
[tex]\[ \text{Amplitude} = \frac{8 - 6}{2} = 1. \][/tex]
Hence, [tex]\( a = 1 \)[/tex].
3. Vertical Shift: The function oscillates around its midline. The midline can be computed as the average of the maximum and minimum values:
[tex]\[ \text{Midline} = \frac{8 + 6}{2} = 7. \][/tex]
So, [tex]\( d = 7 \)[/tex].
4. Horizontal Shift: Finally, the function appears to be a standard sinusoidal wave without a horizontal phase shift, fitting the standard sine wave well at [tex]\(x = 0\)[/tex], suggesting [tex]\( c = 0 \)[/tex].
### Step 4: Writing the Equation
Combining all these parameters, we write the equation, adjusting for the variables identified:
[tex]\[ y = a \sin(bx + c) + d \][/tex]
Substituting the values:
[tex]\[ y = 1 \sin(1x + 0) + 7 \][/tex]
or simplified,
[tex]\[ y = \sin(x) + 7 \][/tex]
### Summary
We identified the parameters [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] from the given periodic points and formed the sinusoidal function:
[tex]\[ y = \sin(x) + 7 \][/tex]
Here, [tex]\(a = 1\)[/tex] represents the amplitude, [tex]\(k = b = 1\)[/tex] represents the frequency, and [tex]\(d = 7\)[/tex] represents the vertical shift (midline). No horizontal shift ([tex]\(c = 0\)[/tex]) is necessary for the described points.
Graphing this function will match the given points and help us understand the pattern of the gum's height over time.
### Step 1: Plotting the Points
First, let's plot the given points on a coordinate plane. The points are:
- [tex]\( (0, 6) \)[/tex]
- [tex]\( \left(\frac{\pi}{2}, 7\right) \)[/tex]
- [tex]\( (\pi, 8) \)[/tex]
- [tex]\( \left(\frac{3\pi}{2}, 7\right) \)[/tex]
- [tex]\( (2\pi, 6) \)[/tex]
### Step 2: Observing the Pattern
Examining the plotted points, we notice the following pattern:
1. The function has a periodic nature with points repeating approximately every [tex]\(2\pi\)[/tex].
2. The values oscillate around a central value, peaking, and then dipping in a consistent pattern.
### Step 3: Identifying Parameters
Next, identify the parameters of a sinusoidal function. We consider a function of the form:
[tex]\[ y = a \sin(bx + c) + d \][/tex]
From the given points, we can make the following observations:
1. Periodicity: The function completes one cycle over [tex]\(2\pi\)[/tex], suggesting the period [tex]\(T = 2\pi\)[/tex]. The frequency [tex]\(b\)[/tex] then is
[tex]\[ b = \frac{2\pi}{T} = \frac{2\pi}{2\pi} = 1. \][/tex]
2. Amplitude: The height of the peaks and troughs about the midline of the function can be observed. From the points, the maximum value is 8, and the minimum value is 6:
[tex]\[ \text{Amplitude} = \frac{8 - 6}{2} = 1. \][/tex]
Hence, [tex]\( a = 1 \)[/tex].
3. Vertical Shift: The function oscillates around its midline. The midline can be computed as the average of the maximum and minimum values:
[tex]\[ \text{Midline} = \frac{8 + 6}{2} = 7. \][/tex]
So, [tex]\( d = 7 \)[/tex].
4. Horizontal Shift: Finally, the function appears to be a standard sinusoidal wave without a horizontal phase shift, fitting the standard sine wave well at [tex]\(x = 0\)[/tex], suggesting [tex]\( c = 0 \)[/tex].
### Step 4: Writing the Equation
Combining all these parameters, we write the equation, adjusting for the variables identified:
[tex]\[ y = a \sin(bx + c) + d \][/tex]
Substituting the values:
[tex]\[ y = 1 \sin(1x + 0) + 7 \][/tex]
or simplified,
[tex]\[ y = \sin(x) + 7 \][/tex]
### Summary
We identified the parameters [tex]\(a\)[/tex], [tex]\(b\)[/tex], [tex]\(c\)[/tex], and [tex]\(d\)[/tex] from the given periodic points and formed the sinusoidal function:
[tex]\[ y = \sin(x) + 7 \][/tex]
Here, [tex]\(a = 1\)[/tex] represents the amplitude, [tex]\(k = b = 1\)[/tex] represents the frequency, and [tex]\(d = 7\)[/tex] represents the vertical shift (midline). No horizontal shift ([tex]\(c = 0\)[/tex]) is necessary for the described points.
Graphing this function will match the given points and help us understand the pattern of the gum's height over time.
