Problem 54 - A Graphical Approach to Precalculus with Limits

Question

Find the (a) amplitude, (b) period, (c) phase shift (if any). (d) vertical translation (if any), and (e) range of each finction. Then graph the function over at least one period. $$y=4-3 \cos (x-\pi)$$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \(2\pi\), Phase Shift: \(\pi\) right, Vertical Translation: 4 up, Range: [1, 7].

Step 1: Identify the Amplitude

The amplitude of a cosine function of the form \(y = a \cos(bx + c) + d\) is the absolute value of the coefficient \(a\). Therefore, for the function \(y = 4 - 3 \cos(x - \pi)\), we first rewrite it as \(y = -3 \cos(x - \pi) + 4\). The amplitude is \(|-3| = 3\).

Step 2: Determine the Period

The period of the cosine function is given by \(\frac{2\pi}{b}\), where \(b\) is the coefficient of \(x\) in \(\cos(bx + c)\). In this function, \(b = 1\), so the period is \(\frac{2\pi}{1} = 2\pi\).

Step 3: Calculate the Phase Shift

To find the phase shift, we look for the horizontal shift given by \(-\frac{c}{b}\). In \(y = 4 - 3 \cos(x - \pi)\), \(c = -\pi\) (as it appears to the right of \(x\)), and \(b = 1\). So the phase shift is \(-\frac{-\pi}{1} = \pi\). This means the graph shifts \(\pi\) units to the right.

Step 4: Identify the Vertical Translation

The vertical translation of the function is the constant term \(d\) in \(y = a \cos(bx + c) + d\). For this function, \(d = 4\), which means the entire graph is shifted 4 units upward.

Step 5: Determine the Range

The range of a cosine function is given by \([d - |a|, d + |a|]\). In this function, \(a = -3\) and \(d = 4\). Thus the range is \([4 - 3, 4 + 3] = [1, 7]\). This means the function oscillates between 1 and 7.

Key Concepts

Amplitude

Amplitude is a key feature of trigonometric functions like sine and cosine. It indicates the height from the center line of the wave to its peak. In simpler terms, it tells you how tall the wave is. For the function

• \(y = 4 - 3 \cos(x - \pi)\),
• we can rewrite it as \(y = -3 \cos(x - \pi) + 4\).

Notice the coefficient in front of the cosine, which is

• \(-3\).
• The amplitude is the absolute value of this coefficient,
• so we have \(|-3| = 3\).

This tells us that the wave will rise and fall 3 units from its central horizontal line, making the wave's height 6 units total (3 up and 3 down from the central position). Understanding amplitude is crucial because it affects how high or low the peaks and troughs of the wave reach.

Period

The period of a trigonometric function describes how long it takes for the wave to repeat itself. Think of it as the distance for one complete wave cycle.

• For the cosine function in the exercise, the formula to find the period is \(\frac{2\pi}{b}\).
• Here, \(b\) is the coefficient of the \(x\) term inside the cosine, in \(\cos(bx + c)\).

In our equation

• \(b = 1\),
• so the period becomes \(\frac{2\pi}{1} = 2\pi\).

This means the wave takes a horizontal distance of \(2\pi\) units to complete one full cycle. Understanding the period is essential as it helps visualize how stretched or compressed the wave is along the x-axis.

Phase Shift

Phase shift refers to the horizontal movement of the wave along the x-axis. This tells us whether the graph slides to the right or left compared to the standard cosine graph.

• To find the phase shift, use the formula \(-\frac{c}{b}\).
• For our function, \(c = -\pi\)
• and \(b = 1\).

Thus, the phase shift is

• \(-\frac{-\pi}{1} = \pi\).

This signifies that the entire cosine wave shifts \(\pi\) units to the right. The phase shift is crucial as it affects the starting point of the wave, determining where the wave's peaks and troughs align with the x-axis.

Vertical Translation

Vertical translation is about moving the entire graph up or down along the y-axis. It's represented by the constant term \(d\) in the cosine function form \(y = a \cos(bx + c) + d\).

• In our example, \(d = 4\).

This means the whole graph is shifted 4 units upward.

• Vertical translation affects the middle line around which the wave oscillates.
• For this function, the wave oscillates around the line \(y = 4\).

Understanding vertical translation helps to determine the average value of the wave, which is important for interpreting and graphing the function.

Range of Function

The range of a trigonometric function describes all the possible values the function can take on the y-axis. It tells you how high or low the wave can go.

• For the cosine function given, we calculate the range using \([d - |a|, d + |a|]\),
• where \(a\) is the amplitude coefficient and \(d\) is the vertical translation.

In the function

• \(a = -3\) and
• \(d = 4\).

So,

• \(d - |a| = 4 - 3 = 1\) and
• \(d + |a| = 4 + 3 = 7\).

Therefore, the range is

• \([1, 7]\).
• This means the graph goes no lower than 1 and no higher than 7 on the y-axis.

Grasping the range is crucial since it helps to visualize how tall and how low the graph reaches, giving a full perspective of the function's behavior.