Problem 114 - Algebra and Trigonometry

Question

Determine the amplitude and period of \(y=3 \cos 2 \pi x\) Then graph the function for \(-4 \leq x \leq 4\) (Section \(5.5,\) Example 5 )

Step-by-Step Solution

Answer

The amplitude of the function \(y=3 \cos 2 \pi x\) is 3, and the period is 1. The function will have four complete waves between x=-4 and x=4, each wave reaching a maximum of 3 and minimum of -3.

Step 1: Identify the Amplitude

The amplitude of a function \(y = A \cos(Bx + C) + D\) is the absolute value of A. In this function, \(y=3 \cos 2 \pi x\), A=3. Therefore, the amplitude of the function is 3.

Step 2: Identify the Period

In the equation given, B=2π is the part that determines the period. The period (P) can be found by calculating \(P=\frac{2\pi}{B}\). So, for this equation, \(P=\frac{2\pi}{2\pi}=1\). Therefore, the period of the function is 1.

Step 3: Plotting the Function

Plot the function \(y = 3\cos(2\pi x)\) from x=-4 to x=4. As the period is 1, there will be 4 complete waves (from \(-4 \leq x \leq 4)\) in the graph. The amplitude tells us that the height of the waves will be 3. The waves should oscillate between -3 and 3 on the y-axis. Start by plotting the point (0,3), then continue to complete a full wave every interval of 1 along the x-axis, switching between the peak (amplitude of 3), the equilibrium (y=0), and the trough (amplitude of -3).

Key Concepts

Graphing Trigonometric Functions

Graphing trigonometric functions involves understanding their distinctive wave-like patterns along the coordinate grid. The equation of the form \(y = A \cos(Bx + C) + D\) is a standard cosine function, where each parameter modifies the graph's shape.

Here’s how it breaks down:

• **\(A\)** affects the **amplitude**, determining how high and low the peaks and troughs of the cosine wave go.
• **\(B\)** influences the **period**, telling us how frequently the wave pattern repeats itself.
• **\(C\)** results in a **horizontal shift** of the entire graph.
• **\(D\)** moves the function **vertically**, shifting the base line up or down.

For the function \(y = 3\cos(2\pi x)\), it's easier to graph once we calculate these values. Begin by plotting points using amplitude and period, then connect them smoothly with the cosine curve. This involves periodic peaks and troughs based on the calculated values.

Amplitude

Amplitude is central to understanding the vertical height of trigonometric functions like sine and cosine. When we talk about amplitude, we're discussing the graph's highest and lowest points away from the horizontal axis.

In the given function \(y = 3 \cos 2\pi x\), the amplitude is derived from the coefficient of the cosine term. Thus:

• The amplitude is the absolute value of the coefficient \(A\), which is **3** in this instance.
• This means that the graph will peak at \(y = 3\) and trough at \(y = -3\).

The significance of amplitude is that it determines how tall the waves of the function are. Changing the amplitude will stretch or shrink the graph vertically. You focus on this when you need to modify how intense the wave peaks are.

Period of a Cosine Function

The period of a trigonometric function like cosine determines how often its wave pattern repeats over the x-axis. It's calculated using the formula:

• \(P = \frac{2\pi}{B}\)

For the function \(y = 3 \cos 2\pi x\), \(B = 2\pi\), so:

• The period is \(P = \frac{2\pi}{2\pi} = 1\).

This means that the entire wave pattern will repeat every 1 unit on the x-axis. Consequently, for the range \(-4 \leq x \leq 4\), there are four complete cycles visible. Recognizing the period is crucial because it defines how compressed or expanded the wave will appear horizontally. Adjustments to \(B\) let you control how rapidly or slowly the oscillations occur along the graph.