Problem 90 - A Graphical Approach to Precalculus with Limits

Question

The monthly average temperatures in degrees Fahrenheit at Austin, Texas, are given by \(f(x)=17.5 \sin \left[\frac{\pi}{6}(x-4)\right]+67.5,\) where \(x\) is the month and \(x=1\) corresponds to January. (Source: A. Miller and J. Thompson.) (a) Find the amplitude, period, phase shift, and vertical shift. (b) Determine the maximum and minimum monthly average temperatures and the months when they occur. (c) Make a conjecture as to how the yearly average temperature might be related to \(f(x)\)

Step-by-Step Solution

Answer

(a) Amplitude=17.5, Period=12, Phase shift=4, Vertical shift=67.5. (b) Max temp=85°F in October, Min temp=50°F in April. (c) Yearly average temp is 67.5°F.

Step 1: Identify the General Form

The given function is of the form \( f(x) = a \sin(b(x - c)) + d \). By comparing this with the given function \( f(x)=17.5 \sin \left[\frac{\pi}{6}(x-4)\right]+67.5 \), we identify: - \( a = 17.5 \) (coefficient of the sine function) - \( b = \frac{\pi}{6} \) (coefficient of \( x \)) - \( c = 4 \) (phase shift) - \( d = 67.5 \) (vertical shift)

Step 2: Calculate Amplitude

Amplitude is the absolute value of \( a \). So, the amplitude of \( f(x) \) is: \(|a| = |17.5| = 17.5\)

Step 3: Calculate Period

The period of the sinusoidal function \( \sin(bx) \) is given by \( \frac{2\pi}{b} \). Hence, \[\text{Period} = \frac{2\pi}{\frac{\pi}{6}} = 12\]

Step 4: Determine Phase Shift

The phase shift is represented by \( c \). Since the function has \( (x-4) \), the phase shift towards the right is 4 units.

Step 5: Determine Vertical Shift

The vertical shift of the function is \( d \). Thus, the graph is shifted upwards by 67.5 units.

Step 6: Analyze Maximum and Minimum Temperatures

The maximum value of the sine function is 1 and the minimum is -1. Therefore: - Max temperature = \( 17.5(1) + 67.5 = 85\) - Min temperature = \( 17.5(-1) + 67.5 = 50\) To find months, use the phase shift and period. - Maximum temperature occurs when \( \sin(\frac{\pi}{6}(x-4)) = 1 \), which happens at the peak of the sine function periods (e.g., \(x = 10\) for October). - Minimum temperature occurs when \( \sin(\frac{\pi}{6}(x-4)) = -1 \), which happens at \(x = 4\) (April).

Step 7: Connect Yearly Average Temperature

The sum of maximum and minimum values divided by 2 gives an idea about the average over a period. Thus, the yearly average temperature might closely follow the vertical shift, which is 67.5. Therefore, due to symmetry of the sine function, average \[ \text{Yearly average} = 67.5\]

Key Concepts

Amplitude

When discussing sinusoidal functions like the one representing average temperature in Austin, Texas, amplitude is a crucial concept. Amplitude represents the height of the peaks or depth of the valleys from the function's midline.
It is always a positive number and is calculated as the absolute value of the coefficient of the sine function.

In our function, the coefficient is 17.5, so we find:

• Amplitude = |17.5| = 17.5

This numeric value signifies how much the temperature oscillates above and below its average level throughout the year. Therefore, the temperatures fluctuate 17.5 degrees above and below the average midline of 67.5°F.

Phase Shift

The phase shift of a sinusoidal function like our temperature function defines how the function is horizontally displaced from its original position.
It ensures that the periodic peaks and troughs are aligned with the actual data.To find the phase shift, we look at the term inside the sine function's parentheses. It is expressed as \(x-c\), where \(c\) indicates the phase shift:

• Phase Shift = 4

This means the sine curve is shifted 4 units to the right, realigning its peaks and valleys to reflect Austin's temperature pattern accurately. This shift ensures that the annual temperature cycle syncs with the calendar year.

Vertical Shift

Vertical shift in a trigonometric function refers to how much the entire function moves up or down along the y-axis.
It adjusts the central position of the sinusoidal function, allowing it to match real-life data more accurately. In our function, the vertical shift is denoted by the constant term added to the sine function, which in this case is 67.5:

• Vertical Shift = 67.5

This number signifies that the entire graph of the temperature data is shifted upwards by 67.5 units. It is equivalent to moving the midline of the function to be at 67.5, thereby centering the temperature oscillations around the typical yearly average temperature.

Period of a Function

The period of a function is a measure of how long it takes for the function to complete one full cycle and begin repeating itself again.
For sinusoidal functions like ours, this is usually defined by the term inside the function's sine component.The period of such functions is calculated by dividing the standard period of sine, which is \(2\pi\), by the coefficient of x inside the parenthesis:\[\text{Period} = \frac{2\pi}{\frac{\pi}{6}} = 12\]

• Period = 12 months

This period tells us that the temperature cycle in Austin, Texas repeats every 12 months, fitting that the data represents monthly average temperatures over a typical year.