Problem 54 - Precalculus: A Concise Course

Question

Sketch the graph of the function. (Include two full periods.) $$ y=-3+5 \cos \frac{\pi t}{12} $$

Step-by-Step Solution

Answer

The graph of the function \(y=-3+5 \cos \frac{\pi t}{12}\) starts at a maxima, then decreases to cross the middle point at half the period, reaches a minima at the beginning of next half period, then rises again to cross the middle point at three-fourth of the period, and finally reaches the maxima at the end of the period, and the pattern repeats for the next period.

Step 1: Identify the amplitude, period, phase shift, and vertical shift

The general form of the cosine function is \(y=A \cos (B(t-C))+D\), where: \n A is the amplitude, \n B determines the period, \n C is the phase shift, \n D is the vertical shift. \nLooking at the function \(-3+5 \cos \frac{\pi t}{12}\), we can identify: \n Amplitude(A) = 5, \n B = \frac{\pi}{12}, which makes the period = \(\frac{2\pi}{B} = 24\), \nThere's no phase shift as there's no \(t-C\) structure, \n The vertical shift(D) = -3.

Step 2: Plot the initial points

Now use this information to start plotting the graph. Begin by plotting the vertical shift of -3, and then plot points corresponding to the period, which is 24. Divide this period into four equal intervals to identify key points of the cosine function, i.e., maximum, zero, minimum, zero, and maximum respectively.

Step 3: Draw the function

Once the key points are plotted, draw the function. It should start at a maximum, go down to zero halfway through the first half period, reach a minimum at the start of the second half period, back to zero halfway through the second half period, and then reach maximum at the end of the period.

Step 4: Sketch two complete periods

We have completed one period. To sketch two full periods, simply repeat the pattern as sketched for another period. This procedure gives us two full periods of the given function.

Key Concepts

Amplitude

The amplitude of a trigonometric function, such as the cosine function, represents the height or the maximum displacement from the central axis of the graph. In the equation given, which is in the form \(y = A \cos(Bt) + D\), the amplitude is represented by the absolute value of \(A\).
For \(-3 + 5 \cos \frac{\pi t}{12}\), the amplitude is 5. This means that the graph of this function oscillates 5 units above and below the central vertical axis, which is determined by the vertical shift.

• The amplitude essentially characterizes how "tall" the wave is.
• It is always a positive number, because it represents a distance.
• Larger amplitudes indicate taller waves, while smaller amplitudes indicate shorter waves.

Understanding amplitude is crucial because it affects how the peaks and valleys of the graph are formed.

Period

The period of a cosine function determines how long it takes to complete one full cycle of the wave from start to finish. For a cosine function in this form, \(y = A \cos(Bt) + D\), the period is calculated as \(\frac{2\pi}{|B|}\).
In the given function \(-3 + 5 \cos \frac{\pi t}{12}\), the value of \(B\) is \(\frac{\pi}{12}\). Thus, the period of this function is \(\frac{2\pi}{\frac{\pi}{12}} = 24\).

• The period indicates how many units along the x-axis it takes for the function to repeat its pattern.
• A period of 24 means that the graph will complete one full cycle every 24 units.
• If you were to sketch the graph, after 24 units, the shape and height of the wave will start over.

It's crucial to know the period as it helps to divide the x-axis into manageable sections where you can expect specific characteristics of the wave.

Cosine Function

The cosine function is a fundamental trigonometric function that is known for its wave-like, periodic behavior. The general shape of a cosine function looks like a set of smooth, repeating waves.
In its standard form, the cosine function is expressed as \(y = A \cos(Bt) + D\), where each component changes the shape or orientation of the wave.

• The function starts at a maximum when there is no phase shift.
• It decreases to zero, reaches a minimum, returns to zero, and goes back to a maximum.
• This makes the cosine function particularly useful in modeling scenarios with regular, repeating patterns, like sound waves or tides.

Recognizing the cosine function's graph allows you to anticipate the behavior of any cosine-based formula just by knowing its amplitude, period, and shift properties.

Vertical Shift

The vertical shift in a cosine function moves the entire graph up or down on the coordinate plane. This is represented by \(D\) in the function \(y = A \cos(Bt) + D\).
In the function given, \(-3 + 5 \cos \frac{\pi t}{12}\), the vertical shift is \(-3\).

• This means that the graph is moved down 3 units from its usual position.
• Without the vertical shift, the cosine function oscillates symmetrically around the x-axis (y=0).
• With a vertical shift of \(-3\), the central axis, which the function oscillates around, is now at \(-3\) instead of 0.

Vertical shifts are crucial in adjusting a graph to better fit real-world data, ensuring that models align more closely with observed phenomena.