Problem 39 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 5-40: Find the amplitude, the period, and the phase shift and sketch the graph of the equation. $$ y=5 \cos (2 x+2 \pi)+2 $$

Step-by-Step Solution

Answer

Amplitude: 5, Period: \(\pi\), Phase Shift: \(-\pi\) left.

Step 1: Identify the General Form

The given equation is \( y = 5 \cos(2x + 2\pi) + 2 \). This is similar to the standard form \( y = a \cos(bx - c) + d \) where \( a \) is the amplitude, \( b \) affects the period, \( c \) determines the phase shift, and \( d \) is the vertical shift.

Step 2: Determine the Amplitude

In the equation \( y = 5 \cos(2x + 2\pi) + 2 \), the amplitude is given by \(|a|\). Here, \( a = 5 \), so the amplitude is \( |5| = 5 \).

Step 3: Calculate the Period

The period of the cosine function is determined by \(\frac{2\pi}{b}\). In this case, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).

Step 4: Find the Phase Shift

To find the phase shift, use \( \frac{c}{b} \) from the equation \( y = a \cos(bx - c) + d \). Here, the equation is written as \( y = 5\cos[2(x + \pi)] + 2 \). Therefore, the phase shift is \( -\pi \) to the left.

Step 5: Sketch the Graph

Start by sketching the basic \( \cos(x) \) graph. Apply the transformations: scale the amplitude by 5, shift \( \pi \) units to the left, adjust for a period of \( \pi \), and move the entire graph 2 units up. This involves plotting peak points at \( y = 7 \) and troughs at \( y = -3 \) with midline at \( y = 2 \).

Key Concepts

Amplitude

In trigonometry, the amplitude of a trigonometric function like the cosine function gives us the height of the wave from the central axis or midline. It is a measure of how far the values of the function can reach above or below this central line. For the cosine function in the standard form \( y = a \cos(bx - c) + d \), the amplitude is determined by the value of \( |a| \).
In our given equation, \( y = 5 \cos(2x + 2\pi) + 2 \), the amplitude is \( |5| = 5 \). This means the graph of the function will oscillate 5 units above and below the midline. It's important to note that amplitude is always a positive value, even if the value of \( a \) itself is negative.
This amplitude tells us that the maximum value of the cosine wave will be 5 units above the midline \( y = 2 \), resulting in a maximum point of \( y = 7 \), while the minimum value will be 5 units below the midline, at \( y = -3 \).
This stretching of the graph vertically does not affect the horizontal features such as the period or phase shift, but it changes how high and low the peaks and troughs of the wave go.

Period

The period of a trigonometric function defines how often the function repeats itself. For our cosine function, the period is determined by the coefficient \( b \) in front of \( x \) in the expression \( y = a \cos(bx - c) + d \).
Here's the formula:

• Period = \( \frac{2\pi}{b} \)

In our equation \( y = 5 \cos(2x + 2\pi) + 2 \), the value of \( b \) is 2, so the period is \( \frac{2\pi}{2} = \pi \).
This means that the cosine wave will complete one full cycle over an interval of length \( \pi \). This compression of the wave along the x-axis makes it repeat faster than the standard cosine function, which has a period of \( 2\pi \).
This feature is crucial in accurately sketching the graph, as it determines how frequently the peaks and valleys occur as the x-value increases.

Phase Shift

Phase shift describes the horizontal movement of the graph of a trigonometric function along the x-axis. In our context, it tells us where the cycle of the cosine wave starts compared to the standard \( \cos(x) \) function.
To identify the phase shift from \( y = a \cos(bx - c) + d \) form, we employ:

• Phase Shift = \( \frac{c}{b} \)

Given the equation \( y = 5 \cos[2(x + \pi)] + 2 \), which matches the form of a cosine function, we find that our phase shift is \( -\pi \).
This indicates that the entire graph will be shifted \( \pi \) units to the left. In simpler words, the start point of the first cycle will occur earlier than it would without this shift. Phase shifts can shift the graph either to the left (negative shift) or to the right (positive shift), depending on the value of \( c \).
Knowing the phase shift is essential as it adjusts the positioning of the wave, aligning the cosine graph correctly on the x-axis for graphing purposes.

Graph of Cosine Function

Graphing the cosine function involves considering all transformations, including amplitude, period, and phase shift, which all affect the graphical representation. For our function \( y = 5 \cos(2x + 2\pi) + 2 \), the process will include several clear steps.
To sketch the graph:

• Start with the standard \( \cos(x) \) graph, which oscillates between 1 and -1.
• Adjust the amplitude to 5, which stretches the graph vertically to oscillate between 7 and -3, centered about the midline at \( y = 2 \).
• Modify the period from \( 2\pi \) to \( \pi \), resulting in each full oscillation occurring over an interval of \( \pi \).
• Shift the entire graph \( \pi \) units to the left to account for the phase shift.
• Finally, translate the graph vertically by 2 units upward.

These transformations together shape the final graph of the cosine wave, ensuring it reflects all the changes accounted from the original components. This graphical representation plays a crucial role in helping visualize the effects of each parameter, such as changes in wavelength, peak position, and overall wave behavior across a set interval.