Problem 28 - 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. $$ y=-4 \cos \left(2 x+\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

Amplitude: 4, Period: \(\pi\), Phase shift: \(-\frac{\pi}{6}\).

Step 1: Identify Amplitude

The amplitude of a function in the form \( y = a \cos(bx + c) \) is the absolute value of the coefficient \( a \). Here, \( a = -4 \). The amplitude is \(|-4| = 4\). This affects the height of the wave.

Step 2: Determine the Period

The period of the cosine function \( y = a \cos(bx + c) \) is determined by the formula \( \frac{2\pi}{b} \). Here, \( b = 2 \), so the period is \( \frac{2\pi}{2} = \pi \). This indicates how long it takes for the wave to complete one cycle.

Step 3: Calculate the Phase Shift

The phase shift is found using the expression \( -\frac{c}{b} \) for \( y = a \cos(bx + c) \). Here, \( c = \frac{\pi}{3} \) and \( b = 2 \), so the phase shift is \( -\frac{\frac{\pi}{3}}{2} = -\frac{\pi}{6} \). This represents a shift to the left by \( \frac{\pi}{6} \) units.

Step 4: Sketch the Graph

Begin by plotting the standard cosine wave over one period of \( \pi \). Amplitude will stretch the wave vertically to a peak at \( 4 \) and a trough at \(-4\). Then, shift the entire graph to the left by \( \frac{\pi}{6} \) to account for the phase shift. Repeat the pattern to complete at least one full cycle.

Key Concepts

Amplitude

Understanding the amplitude of a trigonometric function like the cosine function is crucial for graphing. Amplitude refers to the height of the wave, essentially how "tall" the peaks and how "deep" the troughs are. In a function of the form \( y = a \cos(bx + c) \), the amplitude is the absolute value of \( a \), which means we ignore any negative sign if present.

• In our given equation \( y = -4 \cos\left(2x + \frac{\pi}{3}\right) \), the coefficient \( a \) is \(-4\).
• Therefore, the amplitude is \(|-4| = 4\).

This means the maximum value the wave reaches is 4 units above and below the central axis.
An easy way to visualize amplitude is by imagining ropes swaying side to side; the sway represents the amplitude. Remember, in graphing, amplitude stretches or compresses the wave vertically but does not affect its horizontal length.

Period

The period of a trigonometric function determines how long it takes for the wave to complete one full cycle. This is where you begin and end, seeing the same wave shape start repeating. For the function \( y = a \cos(bx + c) \), the period is calculated using the expression \( \frac{2\pi}{b} \).

• In the equation \( y = -4 \cos\left(2x + \frac{\pi}{3}\right) \), \( b = 2 \).
• This gives us a period of \( \frac{2\pi}{2} = \pi \).

This means after every \( \pi \) units along the x-axis, the wave will start repeating its pattern.
Imagine you are tracing the wave with your finger; starting from any point, once you reach the same point on the wave again, you've covered one full period. Adjusting \( b \) changes how quickly these repetitions happen, compressing or expanding the wave horizontally.

Phase Shift

The phase shift determines the horizontal movement of the trigonometric function, shifting the graph either left or right along the x-axis. It answers the question: where does the cycle begin? The phase shift is calculated using \(-\frac{c}{b}\) in the function form \( y = a \cos(bx + c) \).

• In our equation \( c = \frac{\pi}{3} \) and \( b = 2 \).
• This means the phase shift is \(-\frac{\frac{\pi}{3}}{2} = -\frac{\pi}{6}\).

Our phase shift tells us the wave shifts \( \frac{\pi}{6} \) units to the left.
Visualize the phase shift like adjusting the starting point of a swing; it doesn't change the shape or size of the swing, just where it starts in its back-and-forth motion.

Cosine Function

The cosine function is one of the basic building blocks in trigonometry. It's represented by \( y = \cos(x) \) as its simplest form, producing a wave-like pattern when graphed. This function is periodic, meaning it repeats at regular intervals.

• The standard cosine wave starts at its maximum value of 1, decreases to -1, and returns.
• This creates a smooth, continuous wave or "curve."

In the standard unit circle, the cosine value corresponds to the x-coordinate of a point on the circle.
When analyzing equations like \( y = -4 \cos\left(2x + \frac{\pi}{3}\right) \), you're looking at a transformation of the base cosine graph. The changes in amplitude, period, and phase shift add flexibility to how the cosine curve is displayed.
Always remember, trigonometric functions like cosine allow us to describe oscillations and waves mathematically, applicable in various real-world scenarios from sound waves to engineering structures.