Question
City Name:
Latitude of the City:
3
Mar
4
Apr
5
May
6
Jun
Average
Temperature
Month 1
2
Jan
Feb
Month 7
Jul
8
Aug
9
Sep
10
Oct
11
Nov
12
Dec
Average
Temperature
Sources
© 2016 Connections Education LLC. All rights reserved.
Step 2
Plot the 12 values on a graph. The number corresponding to the month is the first
coordinate of each point, and average temperature for that month is the second
coordinate. The points should create a periodic pattern. Assume that the data is, in
fact, periodic and use the graph to determine the following values for both a sine
function and a cosine function:
Vertical Shift
Horizontal Shift
Amplitude
Period
Domain
Range
Sine Cosine
Based on the data in your table, write an equation for a sine function and for a
cosine function.
Sine Function
Cosine Function
Latitude of the City:
3
Mar
4
Apr
5
May
6
Jun
Average
Temperature
Month 1
2
Jan
Feb
Month 7
Jul
8
Aug
9
Sep
10
Oct
11
Nov
12
Dec
Average
Temperature
Sources
© 2016 Connections Education LLC. All rights reserved.
Step 2
Plot the 12 values on a graph. The number corresponding to the month is the first
coordinate of each point, and average temperature for that month is the second
coordinate. The points should create a periodic pattern. Assume that the data is, in
fact, periodic and use the graph to determine the following values for both a sine
function and a cosine function:
Vertical Shift
Horizontal Shift
Amplitude
Period
Domain
Range
Sine Cosine
Based on the data in your table, write an equation for a sine function and for a
cosine function.
Sine Function
Cosine Function
Answers
Answered by
GPT-4o mini
To analyze the average temperature data for a city and create sine and cosine functions based on that data, we need to first identify the relevant values and then derive the equations.
### Step 1: Sample Data
Let's assume that we have the following average temperature data for each month:
- January: 30°F
- February: 32°F
- March: 45°F
- April: 55°F
- May: 65°F
- June: 75°F
- July: 80°F
- August: 78°F
- September: 70°F
- October: 55°F
- November: 40°F
- December: 33°F
### Step 2: Plotting the Data
The coordinate points for the graph would be:
1. (1, 30)
2. (2, 32)
3. (3, 45)
4. (4, 55)
5. (5, 65)
6. (6, 75)
7. (7, 80)
8. (8, 78)
9. (9, 70)
10. (10, 55)
11. (11, 40)
12. (12, 33)
### Step 3: Determine Values for Sine and Cosine Functions
Using the data points, we can determine the following characteristics of the periodic functions:
1. **Vertical Shift** (D):
This is the average of the maximum and minimum temperatures.
- Max: 80°F (July)
- Min: 30°F (January)
- Average: \(D = \frac{80 + 30}{2} = 55°F\)
2. **Amplitude** (A):
This is half the distance between the maximum and minimum temperatures.
- \(A = \frac{80 - 30}{2} = 25°F\)
3. **Period** (P):
The period of the temperature cycle is 12 months.
- \(P = 12\)
4. **Domain** (in months):
The function describes the temperatures from January to December, so the domain is [1, 12].
5. **Range**:
The range of the average temperatures is from a minimum of 30°F to a maximum of 80°F.
- Range: [30, 80]
6. **Horizontal Shift**:
For the sine function, there is no horizontal shift, but for the cosine function, since the peak occurs in July (month 7), the horizontal shift can be 7 months to the left.
### Step 4: Sine and Cosine Function Equations
1. **Sine Function**:
The equation is generally modeled as:
\[
T(x) = A \sin\left(\frac{2\pi}{P}(x - \text{Phase})\right) + D
\]
Since the sine function typically starts at the midline:
\[
T(x) = 25 \sin\left(\frac{2\pi}{12}(x - 3)\right) + 55
\]
Here, the phase shift is 3 months (March).
2. **Cosine Function**:
The equation is generally modeled as:
\[
T(x) = A \cos\left(\frac{2\pi}{P}(x - \text{Phase})\right) + D
\]
For the cosine function, we adjust for the maximum peak:
\[
T(x) = 25 \cos\left(\frac{2\pi}{12}(x - 7)\right) + 55
\]
### Summary
- **Sine Function**:
\[
T(x) = 25 \sin\left(\frac{2\pi}{12}(x - 3)\right) + 55
\]
- **Cosine Function**:
\[
T(x) = 25 \cos\left(\frac{2\pi}{12}(x - 7)\right) + 55
\]
These functions describe the average temperature throughout the year, based on the plot that you would have made using the given average temperatures with the provided points.
### Step 1: Sample Data
Let's assume that we have the following average temperature data for each month:
- January: 30°F
- February: 32°F
- March: 45°F
- April: 55°F
- May: 65°F
- June: 75°F
- July: 80°F
- August: 78°F
- September: 70°F
- October: 55°F
- November: 40°F
- December: 33°F
### Step 2: Plotting the Data
The coordinate points for the graph would be:
1. (1, 30)
2. (2, 32)
3. (3, 45)
4. (4, 55)
5. (5, 65)
6. (6, 75)
7. (7, 80)
8. (8, 78)
9. (9, 70)
10. (10, 55)
11. (11, 40)
12. (12, 33)
### Step 3: Determine Values for Sine and Cosine Functions
Using the data points, we can determine the following characteristics of the periodic functions:
1. **Vertical Shift** (D):
This is the average of the maximum and minimum temperatures.
- Max: 80°F (July)
- Min: 30°F (January)
- Average: \(D = \frac{80 + 30}{2} = 55°F\)
2. **Amplitude** (A):
This is half the distance between the maximum and minimum temperatures.
- \(A = \frac{80 - 30}{2} = 25°F\)
3. **Period** (P):
The period of the temperature cycle is 12 months.
- \(P = 12\)
4. **Domain** (in months):
The function describes the temperatures from January to December, so the domain is [1, 12].
5. **Range**:
The range of the average temperatures is from a minimum of 30°F to a maximum of 80°F.
- Range: [30, 80]
6. **Horizontal Shift**:
For the sine function, there is no horizontal shift, but for the cosine function, since the peak occurs in July (month 7), the horizontal shift can be 7 months to the left.
### Step 4: Sine and Cosine Function Equations
1. **Sine Function**:
The equation is generally modeled as:
\[
T(x) = A \sin\left(\frac{2\pi}{P}(x - \text{Phase})\right) + D
\]
Since the sine function typically starts at the midline:
\[
T(x) = 25 \sin\left(\frac{2\pi}{12}(x - 3)\right) + 55
\]
Here, the phase shift is 3 months (March).
2. **Cosine Function**:
The equation is generally modeled as:
\[
T(x) = A \cos\left(\frac{2\pi}{P}(x - \text{Phase})\right) + D
\]
For the cosine function, we adjust for the maximum peak:
\[
T(x) = 25 \cos\left(\frac{2\pi}{12}(x - 7)\right) + 55
\]
### Summary
- **Sine Function**:
\[
T(x) = 25 \sin\left(\frac{2\pi}{12}(x - 3)\right) + 55
\]
- **Cosine Function**:
\[
T(x) = 25 \cos\left(\frac{2\pi}{12}(x - 7)\right) + 55
\]
These functions describe the average temperature throughout the year, based on the plot that you would have made using the given average temperatures with the provided points.