Problem 22 - Calculus Early Transcendentals: Pearson New International Edition

Question

Determine the period, amplitude, and shifts (both horizontal and vertical) and draw a graph over the interval \(-5 \leq x \leq 5\) for the functions listed. $$ y=3 \cos \left(x-\frac{\pi}{2}\right)-1 $$

Step-by-Step Solution

Answer

Amplitude: 3, Period: \(2\pi\), Horizontal shift: \(\frac{\pi}{2}\) right, Vertical shift: 1 down.

Step 1: Identify the Standard Form of Cosine Function

The standard form for a transformed cosine function is \(y = A \cos(B(x - C)) + D\), where \(A\) is the amplitude, \(B\) affects the period, \(C\) is the horizontal shift, and \(D\) is the vertical shift. The function given is \(y = 3 \cos \left(x - \frac{\pi}{2}\right) - 1\).

Step 2: Determine the Amplitude

The amplitude of the function is the absolute value of \(A\). In this function, \(A = 3\). Thus, the amplitude is 3.

Step 3: Calculate the Period

The period of a cosine function is calculated by the formula \(\frac{2\pi}{B}\). In this case, \(B = 1\) because it's the coefficient of \(x\) inside the cosine. Therefore, the period is \(\frac{2\pi}{1} = 2\pi\).

Step 4: Find the Horizontal Shift

The horizontal shift is determined by \(C\) in the expression \(x - C\). Here, \(C = \frac{\pi}{2}\). Thus, there is a horizontal shift to the right by \(\frac{\pi}{2}\).

Step 5: Determine the Vertical Shift

The vertical shift is given by \(D\). Here, \(D = -1\), indicating a vertical shift downward by 1 unit.

Step 6: Draw the Graph

Since we are drawing the graph from \(-5\) to \(5\), plot the cosine wave with the amplitude 3, period \(2\pi\), shifted right by \(\frac{\pi}{2}\), and downward by 1 unit. Mark points like peaks, troughs, and zeros considering these transformations and make the wave repeat based on the period.

Key Concepts

Amplitude

The amplitude of a trigonometric function, such as a sine or cosine function, is a measure of the height of the wave. Simply put, it is the distance from the middle of the wave to its peak. In mathematical terms, for a function following the form \( y = A \cos(B(x - C)) + D \), amplitude is represented by \( |A| \).
For the given function \( y = 3 \cos \left(x - \frac{\pi}{2}\right) - 1 \), the amplitude is calculated as follows:

• The coefficient in front of the cosine function is \( A = 3 \).
• The amplitude, therefore, is \( |A| = 3 \).

This tells us that this particular cosine function will reach a maximum of 3 units above and below its midline, before any vertical shift is considered.

Period

The period of a trigonometric function is the distance over which the function repeats itself. For cosine functions, the standard period is \( 2\pi \). However, transformations can alter this period.
In our function \( y = 3 \cos \left(x - \frac{\pi}{2}\right) - 1 \), the period is affected by the coefficient of \( x \) inside the cosine function, known as \( B \). Let's see how this impacts the period:

• In the given function, \( B = 1 \).
• The formula for calculating the period is \( \frac{2\pi}{B} \).
• As such, the period of our function is \( \frac{2\pi}{1} = 2\pi \).

This means the wave completes one full cycle every \( 2\pi \) units along the x-axis.

Horizontal Shift

The horizontal shift, also known as the phase shift, is the amount by which the function is shifted horizontally on the graph. In simpler terms, it indicates how far the graph has moved right or left from its standard position.
For cosine functions in the form \( y = A \cos(B(x - C)) + D \), \( C \) determines the horizontal shift:

• Our function is \( y = 3 \cos \left(x - \frac{\pi}{2}\right) - 1 \).
• Here, \( C = \frac{\pi}{2} \).
• This results in a horizontal shift to the right by \( \frac{\pi}{2} \) units.

The function's wave, therefore, starts \( \frac{\pi}{2} \) units to the right of where it would normally begin in a standard cosine function.

Vertical Shift

Vertical shift describes how far the entire graph of the function is moved up or down. This transformation makes the function's midline not coincide with the x-axis.
When looking at functions of the form \( y = A \cos(B(x - C)) + D \), \( D \) specifies the vertical shift:

• For our example function, \( y = 3 \cos \left(x - \frac{\pi}{2}\right) - 1 \), \( D = -1 \).
• This translates to a vertical shift downward by 1 unit.

This means the midline of the wave is shifted down by 1 unit, affecting the placement of both the peaks and the troughs of the wave on the graph.