Problem 16 - 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. $$ y=\cos (2 x-\pi)+2 $$

Step-by-Step Solution

Answer

Amplitude: 1; Period: \(\pi\); Phase Shift: \(\frac{\pi}{2}\) to the right.

Step 1: Identify the Amplitude

The general form of a cosine function is \( y = a \cos(bx - c) + d \). Here, \( a \) represents the amplitude. In the given function \( y = \cos(2x - \pi) + 2 \), \( a = 1 \) since it's the coefficient of the cosine function. Therefore, the amplitude is 1.

Step 2: Determine the Period

For a cosine function in the form \( y = a \cos(bx - c) + d \), the period is given by \( \frac{2\pi}{b} \). In this case, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \).

Step 3: Calculate the Phase Shift

The phase shift of the cosine function is calculated using \( \frac{c}{b} \), where \( c \) is the horizontal shift and \( b \) is the coefficient of \( x \). Here, \( c = \pi \) and \( b = 2 \), so the phase shift is \( \frac{\pi}{2} \) to the right.

Step 4: Determine the Vertical Shift

The vertical shift \( d \) is the constant added outside the cosine function. For \( y = \cos(2x - \pi) + 2 \), \( d = 2 \). This means the entire graph shifts upwards by 2 units.

Step 5: Sketch the Graph

Start by sketching the basic cosine function \( \cos(x) \). Adjust the amplitude to 1 (no change), the period to \( \pi \) by compressing the wave horizontally, apply a phase shift of \( \frac{\pi}{2} \) to the right, and move the entire graph up by 2 units to account for the vertical shift.

Key Concepts

Amplitude of a Function

The amplitude of a cosine function is a measure of how much the wave stretches vertically. Think of it as half the distance between the highest and lowest points of the wave. This can be visualized as the height from the centerline of the wave to a peak or a trough.
In the general cosine function format, \( y = a \cos(bx - c) + d \), the amplitude is represented by \( |a| \).
For the equation \( y = \cos(2x - \pi) + 2 \), the amplitude is the coefficient of the cosine function, which is 1.

• The amplitude tells us the maximum height of the wave above and below the centerline.
• It defines the peak strength or intensity of the wave.

With an amplitude of 1, the cosine wave maintains its standard height, ensuring the characteristic smooth, wave-like oscillations are preserved.

Period of a Trigonometric Function

A trigonometric function’s period is the horizontal length of one complete cycle of the wave. It tells us how long it takes for the function to return to its starting point.
For the cosine function \( y = a \cos(bx - c) + d \), the period is calculated using the formula \( \frac{2\pi}{b} \).
In our function \( y = \cos(2x - \pi) + 2 \), \( b \) is 2, so the period becomes \( \frac{2\pi}{2} = \pi \).

• The period of \( \pi \) means the wave completes a full cycle every \( \pi \) units on the x-axis.
• This compressed period results from the value of \( b = 2 \), indicating a quicker oscillation compared to the standard cosine function which has a period of \( 2\pi \).

The shorter period shows that the wave oscillates twice as fast as the normal cosine wave.

Phase Shift

The phase shift of a trigonometric function describes a horizontal movement along the x-axis. It's essentially the starting point of the wave cycle on the x-axis.
To find the phase shift, we use the formula \( \frac{c}{b} \), derived from the cosine function \( y = a \cos(bx - c) + d \).
For \( y = \cos(2x - \pi) + 2 \), \( c = \pi \) and \( b = 2 \), giving us a phase shift of \( \frac{\pi}{2} \) to the right.

• This shift moves the entire cosine wave \( \frac{\pi}{2} \) units along the x-axis.
• It alters when the wave starts oscillating, essentially moving its starting point.

The rightward shift of \( \frac{\pi}{2} \) adjusts the position of the peaks and troughs of the wave to the right, compared to its standard position.

Vertical Shift

The vertical shift of a function is about moving the whole graph up or down along the y-axis. It's like adjusting the centerline of the wave.
In the cosine function \( y = a \cos(bx - c) + d \), the vertical shift is determined by \( d \).
For \( y = \cos(2x - \pi) + 2 \), the constant \( d = 2 \) signifies a vertical shift upwards by 2 units.

• This means the entire wave, with its peaks and troughs, is lifted two units higher.
• The shift affects all y-values equally, so the oscillations move with the centerline elevated to y = 2.

The upward shift modifies the wave's balance point and affects how you measure the height of the wave, as it doesn't oscillate around the x-axis anymore.