Problem 9 - Precalculus: Functions and Graphs

Question

Find the amplitude, the period, and the phase shift and sketch the graph of the equation. \(y=\cos \left(x+\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

Amplitude: 1, Period: \(2\pi\), Phase Shift: \(-\frac{\pi}{2}\). Graph is shifted left by \(\frac{\pi}{2}\).

Step 1: Identify the General Form

The general form of a cosine function is \(y = a \cos(b(x - c)) + d\). Here, \(a\) is the amplitude, \(b\) affects the period, \(c\) is the phase shift, and \(d\) is the vertical shift. Our equation is \(y = \cos\left(x + \frac{\pi}{2}\right)\), which can be rewritten as \(y = \cos\left(1(x - (-\frac{\pi}{2}))\right)\).

Step 2: Determine the Amplitude

In the equation \(y = \cos\left(1(x - (-\frac{\pi}{2}))\right)\), the amplitude is given by \(|a|\). Since \(a = 1\) (the coefficient of \(\cos\) term), the amplitude is \(1\).

Step 3: Calculate the Period

The period of a cosine function is calculated with the formula \(\frac{2\pi}{b}\). Here, \(b = 1\), so the period is \(\frac{2\pi}{1} = 2\pi\).

Step 4: Determine the Phase Shift

The phase shift is determined by the value \(c\) in the function \(y = \cos(b(x - c))\). Our equation can be rewritten from \(y = \cos(x + \frac{\pi}{2})\) to \(y = \cos(x - (-\frac{\pi}{2}))\). Thus, the phase shift is \(-\frac{\pi}{2}\), indicating a shift to the left by \(\frac{\pi}{2}\).

Step 5: Sketch the Graph

To sketch the graph of \(y = \cos\left(x + \frac{\pi}{2}\right)\), start by drawing the standard cosine graph with an amplitude of 1. Shift the entire graph to the left by \(\frac{\pi}{2}\) units. The period remains \(2\pi\), so mark one full cycle from \(-\frac{\pi}{2}\) to \(\frac{3\pi}{2}\) on the x-axis. Ensure the midline remains at \(y=0\), as there is no vertical shift (\(d = 0\)).

Key Concepts

Amplitude

Amplitude is a measure of how far the peaks and valleys of a trigonometric function like the cosine function spread from its equilibrium position, often known as the midline. It tells us how tall or deep the waves go.
In the equation given in the exercise, which is written as: \[ y = a \cos(b(x - c)) + d \] The amplitude is given by \(|a|\). Here, the coefficient \(a\) is \(1\), meaning the amplitude is \(1\). This indicates that the highest point on the graph reaches \(1\) and the lowest point reaches \(-1\), centered around the midline, which is \(y = 0\).
No matter how the graph shifts or stretches, this amplitude keeps the wave height consistent, always centered around its mid-line.

Period

The period of a trigonometric function, like a cosine function, indicates how long it takes for the function to complete one full cycle.
In mathematical terms, it's derived using the formula: \[ \text{Period} = \frac{2\pi}{b} \] where \(b\) affects how fast the wave goes through its cycle. In our exercise, since \(b = 1\), the period remains \(2\pi\). - This period of \(2\pi\) tells us that the wave repeats every \(2\pi\) units of \(x\).
- For this specific cosine function, the feature of completing one cycle in \(2\pi\) units doesn’t change, showing the regularity in its waviness.

Phase Shift

Phase shift involves moving the entire graph of a trigonometric function like a cosine function left or right along the x-axis. It's determined by the value of \(c\) in the transformed function: \[ y = a \cos(b(x - c)) + d \] In our function, expressed as \(y = \cos(x + \frac{\pi}{2})\), it can be rewritten to highlight the phase as \(y = \cos(x - (-\frac{\pi}{2}))\). This rearrangement shows that \(c = -\frac{\pi}{2}\). - Represented as a shift to the left by \(\frac{\pi}{2}\).
This means the original start of a cycle, typically at \(x = 0\) for a standard cosine function, begins instead at \(-\frac{\pi}{2}\).
- This leftward adjustment influences the starting point of peaks and valleys on the graph.

Graph of Trigonometric Functions

Graphing trigonometric functions involves visualizing how these functions behave over a range of values.
For a cosine function like \(y = \cos(x + \frac{\pi}{2})\), several crucial steps are needed:

• First, understand the effect of the amplitude: In our case, it is \(1\), meaning the peaks hit \(y = 1\) and troughs dip to \(y = -1\).
• Next, account for the phase shift: The graph of the cosine function shifts left by \(\frac{\pi}{2}\), which alters the traditional start of its cycle.
• Finally, affirm the period: Since the period is \(2\pi\), the full cycle composes itself over \([-\frac{\pi}{2}, \frac{3\pi}{2}]\) on the x-axis.

When plotting, keep the midline at \(y = 0\) and respect these characteristics' influence as they ensure accuracy in how the function graph appears, reflecting its true mathematical behavior.