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Sketch a graph of the function f(x) = 3 cos 6 5 4 3 2 1 -8п -7п -бп -5п -4п -3л -2л -π -1 -2 -3 -4 -5 -6 Clear All Draw: ΑΛΛΑ 1 -X 3 π 2π 3π 4π 5TT бл 7π 8π

Sketch a graph of the function f(x) = 3 cos 6 5 4 3 2 1 -8п -7п -бп -5п -4п -3л -2л -π -1 -2 -3 -4 -5 -6 Clear All Draw: ΑΛΛΑ 1 -X 3 π 2π 3π 4π 5TT бл 7π 8π

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.5: Graphs Of Functions
Problem 35E
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### Graphing a Function: \( f(x) = 3 \cos \left( \frac{1}{3} x \right) \) In this exercise, you are asked to sketch a graph of the function \( f(x) = 3 \cos \left( \frac{1}{3} x \right) \). #### Step-by-Step Instructions: 1. **Understanding the Function:** - **Amplitude:** The coefficient 3 in front of the cosine function indicates that the amplitude of the function is 3. This means the maximum value of the function is 3 and the minimum value is -3. - **Frequency:** The \( \frac{1}{3} \) inside the cosine function affects the function's frequency. It means that the period of the function is expanded. Specifically, the period \( P \) of the function \( f(x) = 3 \cos \left( \frac{1}{3} x \right) \) is \( P = \frac{2\pi}{\frac{1}{3}} = 6\pi \). 2. **Graph Setup:** - **Horizontal Axis (x-axis):** The horizontal axis is labeled from -8π to 8π in increments of π. - **Vertical Axis (y-axis):** The vertical axis ranges from -6 to 6, with the function oscillating between -3 and 3. 3. **Plotting Points:** - Start by identifying key points at intervals of \( \frac{6\pi}{4} = 1.5\pi \) since this divides the period into quarters. - One complete cycle (from 0 to 6π): - At \( x = 0 \), \( f(0) = 3 \cos(0) = 3 \) - At \( x = 1.5\pi \), \( f(1.5\pi) = 3 \cos(\frac{1.5\pi}{3}) = 3 \cos(0.5\pi) = 3 \times 0 = 0 \) - At \( x = 3\pi \), \( f(3\pi) = 3 \cos(\frac{3\pi}{3}) = 3 \cos(\pi) = 3 \times (-1) = -3 \)
Transcribed Image Text:### Graphing a Function: \( f(x) = 3 \cos \left( \frac{1}{3} x \right) \) In this exercise, you are asked to sketch a graph of the function \( f(x) = 3 \cos \left( \frac{1}{3} x \right) \). #### Step-by-Step Instructions: 1. **Understanding the Function:** - **Amplitude:** The coefficient 3 in front of the cosine function indicates that the amplitude of the function is 3. This means the maximum value of the function is 3 and the minimum value is -3. - **Frequency:** The \( \frac{1}{3} \) inside the cosine function affects the function's frequency. It means that the period of the function is expanded. Specifically, the period \( P \) of the function \( f(x) = 3 \cos \left( \frac{1}{3} x \right) \) is \( P = \frac{2\pi}{\frac{1}{3}} = 6\pi \). 2. **Graph Setup:** - **Horizontal Axis (x-axis):** The horizontal axis is labeled from -8π to 8π in increments of π. - **Vertical Axis (y-axis):** The vertical axis ranges from -6 to 6, with the function oscillating between -3 and 3. 3. **Plotting Points:** - Start by identifying key points at intervals of \( \frac{6\pi}{4} = 1.5\pi \) since this divides the period into quarters. - One complete cycle (from 0 to 6π): - At \( x = 0 \), \( f(0) = 3 \cos(0) = 3 \) - At \( x = 1.5\pi \), \( f(1.5\pi) = 3 \cos(\frac{1.5\pi}{3}) = 3 \cos(0.5\pi) = 3 \times 0 = 0 \) - At \( x = 3\pi \), \( f(3\pi) = 3 \cos(\frac{3\pi}{3}) = 3 \cos(\pi) = 3 \times (-1) = -3 \)
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