Problem 39 - Algebra and Trigonometry

Question

\(29-42\) . Find the amplitude, period, and phase shift of the function, and graph one complete period. $$ y=3 \cos \pi\left(x+\frac{1}{2}\right) $$

Step-by-Step Solution

Answer

The amplitude is 3, the period is 2, and the phase shift is \(-\frac{1}{2}\) units left.

Step 1: Identify the General Form of a Cosine Function

The general form of a cosine function is given by: \[ y = a \cos(b(x - c)) + d \] where \( a \) is the amplitude, \( b \) is related to the period, \( c \) is the phase shift, and \( d \) is the vertical shift. In the given function \( y = 3 \cos \pi\left(x + \frac{1}{2}\right) \), we can compare directly to this form.

Step 2: Determine the Amplitude

The amplitude of a cosine function is the coefficient \( a \) in front of the cosine. In this case, \( a = 3 \). Hence, the amplitude is 3.

Step 3: Calculate the Period

The period of a cosine function is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \) inside the cosine function. Here, \( b = \pi \), so the period is \( \frac{2\pi}{\pi} = 2 \).

Step 4: Find the Phase Shift

To find the phase shift, we need to analyze the expression \( b(x - c) \). In the equation given: \( \pi(x + \frac{1}{2}) \), we can rewrite it as \( \pi(x - (-\frac{1}{2})) \). Thus the phase shift is \( -\frac{1}{2} \), which means the graph is shifted \( \frac{1}{2} \) units to the left.

Step 5: Graph One Complete Period

To graph one complete period, start by marking the phase shift at \( x = -\frac{1}{2} \). The cosine function will start one complete cycle at this point. The period is 2, so the cycle completes at \( x = -\frac{1}{2} + 2 = \frac{3}{2} \). The function will have maximum values (3) and minimum values (-3) because of the amplitude. This results in key points at each quarter of the period, resulting in points at: \(-\frac{1}{2}, 3\), \(0, 0\), \(\frac{1}{2}, -3\), \(1, 0\), and \(\frac{3}{2}, 3\).

Key Concepts

Amplitude

In trigonometric functions, such as the cosine function, the amplitude represents the peak value measured from the midline (or equilibrium position) of the graph. It tells us how "tall" the function is, from the centerline to its highest or lowest point. Let's delve into the specific function provided:

• The function given is \[ y = 3 \cos \pi (x + \frac{1}{2}) \]
• The amplitude is represented by the coefficient in front of the cosine function.
• Here, the coefficient is 3, which means the amplitude is 3.

Therefore, the graph of the cosine function will oscillate 3 units above and 3 units below the central axis of the graph, which is the x-axis in this case. This oscillation is symmetric about the central axis, ensuring that the peaks height is always the amplitude value (3) both positively and negatively.

Period

The period of a trigonometric function expresses how often the function repeats itself. It is the horizontal length of one full cycle of the wave on the graph.To find the period of our cosine function, we use the formula:\[\text{Period} = \frac{2\pi}{b}\]where \( b \) is the coefficient of \( x \) inside the cosine function.For the equation \[ y = 3 \cos \pi (x + \frac{1}{2}) \], the coefficient \( b \) is \( \pi \). Plugging this into the formula we find:

• The period is \[\frac{2\pi}{\pi} = 2\]

This means that every 2 units along the x-axis, the cos function completes one full cycle. As a result, if you looked at the graph from any starting point, exactly two units later, the function would begin to repeat its pattern.

Phase Shift

Phase shift refers to the horizontal shift of a trigonometric graph along the x-axis. It indicates the starting point of the graph relative to the usual position of the base cosine curve. The standard approach to determine the phase shift uses the component \( b(x - c) \) from the general cosine form \( y = a \cos(b(x - c)) + d \). For the function \[ y = 3 \cos \pi (x + \frac{1}{2}) \]:

• The argument is rewritten in the form \( \pi(x - (-\frac{1}{2})) \), revealing \( c = -\frac{1}{2} \).
• This signifies a shift to the left by \(\frac{1}{2}\) units.

In simple terms, the entire graph of the cosine function is moved \( \frac{1}{2} \) units in the negative x-direction. This shift affects where the graph's cycle begins, influencing its intersection points with the x-axis and its symmetry pattern along the grid.