Ex: Model Daily Temperatures Using a Trig Function

Mathispower4u

3 Jul 201207:02

EducationalLearning

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TLDRThe video script discusses modeling daily temperature fluctuations using a sinusoidal function. It starts by establishing the midline at 80 degrees, the average of the day's high (96 degrees) and low (64 degrees) temperatures. The script then uses the given information about the low temperature occurring at 4 AM and the period of 24 hours to plot key points on the graph. By recognizing the graph as a transformation of the cosine function, the video derives the equation for temperature, D(t) = -16 cos(π/12 (t - 4)) + 80, which accurately represents the temperature over a 24-hour period. The explanation is detailed, guiding viewers through each step of the process.

Takeaways

🌡️ The temperature over a day can be modeled with a sinusoidal function.
📈 Given a high temperature of 96 degrees and a low of 64 degrees, the midline of the sinusoidal function is calculated to be 80 degrees.
⏰ The low temperature of 64 degrees occurs at 4 A.M., which helps determine the phase shift in the sinusoidal model.
🔁 A period of 24 hours is established for the sinusoidal function, representing the 24-hour cycle of a day.
📊 The function must contain the points (4,64), (28,64), and (16,96) based on the given temperature data.
📐 The sinusoidal graph is sketched with the midline at y = 80 and reflects a transformation of the basic cosine function.
🔄 The graph is recognized as a reflection across the x-axis, with changes in amplitude, period, and horizontal and vertical shifts.
📉 The amplitude 'a' is determined to be -16, reflecting the graph's transformation and the difference between the high and midline temperatures.
🕒 The period 'b' is calculated as π/12, ensuring the function's period matches the 24-hour cycle.
🔄 The phase shift 'd' is identified as +4, indicating the graph's horizontal shift to the right.
📈 The vertical shift 'c' is +80, aligning the graph's midline with the temperature's average value.

Q & A

What type of function is used to model the outside temperature over a day?
-A sinusoidal function is used to model the outside temperature over a day.
What are the high and low temperatures mentioned in the script?
-The high temperature for the day is 96 degrees, and the low temperature is 64 degrees.
At what time does the low temperature of 64 degrees occur?
-The low temperature of 64 degrees occurs at 4 A.M.
What is the role of the midline in the sinusoidal function model?
-The midline of the sinusoidal function represents the average of the high and low temperatures, which is 80 degrees in this case.
How is the amplitude of the sinusoidal function determined?
-The amplitude is determined by the distance from the midline to the maximum or minimum temperature, which is 16 degrees in this case.
What is the period of the sinusoidal function representing the temperature over a day?
-The period of the sinusoidal function is 24 hours, representing the full day.
What points must the sinusoidal function pass through based on the given information?
-The sinusoidal function must pass through the points (4, 64), (28, 64), and (16, 96).
Why is the cosine function chosen over the sine function for this model?
-The cosine function is chosen because the given data points and the nature of the temperature changes over the day suggest a transformation of the basic cosine function.
How does the reflection across the x-axis affect the value of 'a' in the sinusoidal function?
-The reflection across the x-axis means that 'a' is negative, so 'a' is -16 in this model.
What is the value of 'b' that makes the period of the function 24 hours?
-The value of 'b' is π/12, which makes the period of the function 24 hours.
What are the values of 'd' and 'c' in the sinusoidal function, and what do they represent?
-'d' is +4, representing a rightward shift of the graph by 4 units, and 'c' is +80, representing an upward shift of the graph by 80 units.
What is the final equation of the sinusoidal function in terms of 'D(T)'?
-The final equation of the sinusoidal function in terms of 'D(T)' is D(T) = -16 * COSINE(π/12 * (T - 4)) + 80.

Outlines

00:00

🌡️ Modeling Daily Temperature Variation

This paragraph introduces the concept of modeling daily temperature changes using a sinusoidal function. Given the high temperature of 96 degrees and a low of 64 degrees at 4 A.M., the task is to find an equation for the temperature D in terms of the time T since midnight. The midline is calculated as the average of the high and low temperatures, resulting in 80 degrees. The function's period is determined to be 24 hours, and key points are identified, including (4, 64), (28, 64), and (16, 96). A sinusoidal graph is sketched, and it's recognized as a transformation of the cosine function, reflected across the x-axis with adjusted amplitude, period, and phase shift.

05:01

📉 Determining the Equation of the Sinusoidal Function

The second paragraph focuses on determining the specific values for the sinusoidal function that models the temperature. The amplitude 'A' is found to be -16, reflecting the graph across the x-axis. The period 'B' is calculated to be π/12, which corresponds to a 24-hour cycle. The phase shift 'D' is identified as +4, indicating a rightward shift of the graph. The vertical shift 'C' is determined to be +80, reflecting the midline of the sinusoidal function. The final equation for the temperature D(T) is given as D(T) = -16 * cos(π/12 * (T - 4)) + 80, providing a model for the temperature over a 24-hour period.

Mindmap

Keywords

💡Sinusoidal Function

A sinusoidal function is a mathematical representation of a wave-like pattern that oscillates above and below a central value. In the context of the video, it is used to model the daily temperature fluctuations that occur over a 24-hour period. The script describes how the temperature can be represented by either a sine or cosine function, which are types of sinusoidal functions.

💡Amplitude

Amplitude refers to the maximum displacement from the equilibrium or midline in a wave pattern. In the video, the amplitude is calculated as the difference between the high and low temperatures, which is 16 degrees in this case. It represents the extent of the temperature variation from the average temperature line.

💡Midline

The midline of a sinusoidal function is the horizontal line that represents the average value of the oscillating function. In the script, the midline is determined by averaging the high and low temperatures, resulting in 80 degrees, which serves as the reference point for the temperature fluctuations.

💡Period

The period of a sinusoidal function is the length of one complete cycle of the wave. In the context of the video, the period is 24 hours, which corresponds to the duration of a day. This is used to calculate the value of 'b' in the sinusoidal equation, which determines the frequency of the temperature changes.

💡Phase Shift

A phase shift is a horizontal displacement of the entire wave pattern along the x-axis. In the video, the phase shift is determined by the time at which the low temperature occurs, which is at 4 A.M. This information is used to adjust the sinusoidal equation to match the observed temperature data.

💡Cosine Function

The cosine function is a type of sinusoidal function that starts at its maximum value and decreases to a minimum, then increases back to a maximum, and so on. In the video, the cosine function is chosen to model the temperature fluctuations because the temperature data points align with the characteristics of a cosine wave.

💡Reflection Across the X-axis

Reflection across the x-axis refers to a transformation of the graph of a function where the function's values are mirrored over the x-axis. In the script, the sinusoidal graph is reflected across the x-axis because the temperature pattern starts with a minimum, which is opposite to the natural maximum starting point of the basic cosine function.

💡Horizontal Shift

A horizontal shift is a transformation that moves the graph of a function left or right along the x-axis. The script mentions that the graph has been shifted right by four units, which corresponds to the phase shift needed to align the sinusoidal function with the observed temperature data.

💡Vertical Shift

A vertical shift is a transformation that moves the graph of a function up or down along the y-axis. In the video, the sinusoidal function is shifted up by 80 units to align the midline of the function with the average temperature of 80 degrees.

💡Equation of a Function

The equation of a function is a mathematical formula that defines the relationship between the independent variable (in this case, time) and the dependent variable (temperature). The video script walks through the process of determining the coefficients and transformations needed to create an equation that models the temperature as a function of time.

Highlights

Modeling the outside temperature over a day using a sinusoidal function.

Given high temperature of 96 degrees and low temperature of 64 degrees.

Low temperature occurs at 4 A.M., with T representing hours since midnight.

Finding the midline of the sinusoidal function by averaging high and low temperatures.

The midline is determined to be a horizontal line at y = 80.

Using the points (4,64) and (28,64) to establish the period of the function as 24 hours.

Identifying the point (16,96) where the high temperature occurs.

Plotting the three key points on the coordinate plane.

Recognizing the graph as a transformation of the basic cosine function.

The graph is reflected across the x-axis, indicating a negative amplitude.

Determining the amplitude to be 16, reflected as -16 due to the graph's orientation.

Calculating the period of the function to be 24 hours, leading to B = π/12.

Shifting the graph right by four units, indicating D = +4.

Shifting the graph up by 80 units, resulting in C = +80.

Formulating the sinusoidal function D(T) = -16 * cos(π/12 * (T - 4)) + 80.

The sinusoidal function can model the temperature over a 24-hour period.

Transcripts

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Takeaways

Q & A

What type of function is used to model the outside temperature over a day?

What are the high and low temperatures mentioned in the script?

At what time does the low temperature of 64 degrees occur?

What is the role of the midline in the sinusoidal function model?

How is the amplitude of the sinusoidal function determined?

What is the period of the sinusoidal function representing the temperature over a day?

What points must the sinusoidal function pass through based on the given information?

Why is the cosine function chosen over the sine function for this model?

How does the reflection across the x-axis affect the value of 'a' in the sinusoidal function?

What is the value of 'b' that makes the period of the function 24 hours?

What are the values of 'd' and 'c' in the sinusoidal function, and what do they represent?

What is the final equation of the sinusoidal function in terms of 'D(T)'?