Problem 60 - 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. $$y=-\cos \left[\frac{2}{3}\left(x-\frac{\pi}{3}\right)\right]$$

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(3\pi\), Phase Shift: \(\frac{\pi}{2}\) right, Vertical Translation: none, Range: \([-1, 1]\).

Step 1: Identify the Amplitude

For the function given, the amplitude is the absolute value of the coefficient in front of the cosine function. Since there's no explicit coefficient, the amplitude is 1. Hence the amplitude of the function is 1.

Step 2: Determine the Period

The function is a cosine function of the form \(y = a \cdot \cos(bx + c) + d\). The period of the function is given by \(\frac{2\pi}{|b|}\), where \(b = \frac{2}{3}\). So, the period is \(\frac{2\pi}{\frac{2}{3}} = 3\pi\).

Step 3: Calculate the Phase Shift

To find the phase shift, we look at the term \(b(x - c)\) inside the cosine function. The phase shift is \(-\frac{c}{b}\), which leads to \(-\frac{-\frac{\pi}{3}}{\frac{2}{3}} = \frac{\pi}{2}\) to the right.

Step 4: Determine the Vertical Translation

The general form \(y = a \cdot \cos(bx + c) + d\) shows that \(d\) is the vertical translation. Since \(d = 0\), there is no vertical translation.

Step 5: Find the Range

The range of a cosine function is given by \([-1 \cdot |a| + d, 1 \cdot |a| + d]\). For our function, \(a = -1\) and \(d = 0\). Thus, the range becomes \([-1, 1]\).

Key Concepts

Amplitude

Amplitude in trigonometric functions like sine or cosine describes the height of the wave from its centerline. It tells you how far the peaks and valleys are from the middle of the graph. In the given function, which is a negative cosine function, the formula looks like this:

• Function format: \( y = a \cdot \cos(bx + c) + d \)
• Amplitude is \( |a| \)

In our exercise, the function is \( y = -\cos\left[\frac{2}{3}(x-\frac{\pi}{3})\right] \). There's no number in front of \( \cos \), which means its value is \(-1\). The amplitude is the absolute value, \(|-1| = 1\). The sign does not affect the amplitude, just how the graph is flipped. So, the amplitude is 1 in this function.

Period

The period of a trigonometric function describes how long it takes for the function to repeat its pattern. For cosine and sine functions, the standard period is \(2\pi\).

• To find the period, use \( \frac{2\pi}{|b|} \)

For our function, \( b = \frac{2}{3} \). So the calculation is: \[\text{Period} = \frac{2\pi}{\left| \frac{2}{3} \right|} = 3\pi\]This tells us that the wave pattern will repeat every \( 3\pi \). This impacts how the graph stretches horizontally.

Phase Shift

Phase shift is how the curve shifts horizontally along the x-axis. For the general trigonometric function, the formula to determine phase shift is

• \( -\frac{c}{b} \)

Here, the function given is \(y = -\cos\left[\frac{2}{3}(x-\frac{\pi}{3})\right] \). Compare it with the standard form to find the values of \( b \) and \( c \):

• \( b = \frac{2}{3} \)
• \( c = -\frac{\pi}{3} \)

Calculate the phase shift: \[\text{Phase shift} = -\frac{-\frac{\pi}{3}}{\frac{2}{3}} = \frac{\pi}{2}\]The function shifts to the right by \( \frac{\pi}{2} \). It means the starting point on the graph is moved horizontally.

Vertical Translation

Vertical translation refers to the upward or downward movement of the graph along the y-axis. This is dictated by the \( d \) value in the function equation.

• Format: \( y = a \cdot \cos(bx + c) + d \)

In our problem, since there is no number added or subtracted from the cos function, \( d = 0 \).

• Vertical Translation: None

No vertical translation means the middle line of the wave stays on the x-axis, and there is no shift up or down on the graph.

Range of Function

The range defines the set of output values a function can provide. For cosine functions, the basic output range is from -1 to 1, provided by the maximum and minimum values of the waveform.

• For the function \( y = a \cdot \cos(bx + c) + d \), the range is given by: \[-1 \cdot |a| + d, 1 \cdot |a| + d\]

In the context of our function, where \( a = -1 \) and \( d = 0 \):

• \( \text{Range} = [-1, 1] \)

This means the peaks of this wave hit at -1 and 1 on the y-axis. The entire wave movement is contained within these values, ensuring a consistent oscillation between these limits over the entire graph.