Math Problem Statement

Consider the function f(x) = 3 sin(2x). a. State the amplitude of f(x). b. Determine the period of f(x). c. Sketch the graph of f(x), for 0 ≤ x ≤ 2π.

Solution

Let's solve the problem step by step:

Given Function:

f(x)=3sin(2x)f(x) = 3\sin(2x)

a. Amplitude of f(x)f(x)

The amplitude of a sine function y=asin(bx)y = a\sin(bx) is given by the absolute value of the coefficient aa.

For f(x)=3sin(2x)f(x) = 3\sin(2x), the amplitude is: Amplitude=3=3\text{Amplitude} = |3| = 3

b. Period of f(x)f(x)

The period of a sine function y=asin(bx)y = a\sin(bx) is given by 2πb\frac{2\pi}{b}, where bb is the coefficient of xx inside the sine function.

For f(x)=3sin(2x)f(x) = 3\sin(2x), the coefficient bb is 2. Therefore, the period is: Period=2π2=π\text{Period} = \frac{2\pi}{2} = \pi

c. Sketch the graph of f(x)f(x) for 0x2π0 \leq x \leq 2\pi

To sketch the graph:

  1. The function has an amplitude of 3, so the graph will oscillate between 3-3 and 33.
  2. The period is π\pi, so one complete cycle of the sine wave will occur over the interval [0,π][0, \pi]. Since xx ranges from 00 to 2π2\pi, the graph will show two complete cycles.

Graph Characteristics:

  • At x=0x = 0, f(0)=3sin(0)=0f(0) = 3\sin(0) = 0.
  • At x=π2x = \frac{\pi}{2}, f(π2)=3sin(π)=0f\left(\frac{\pi}{2}\right) = 3\sin(\pi) = 0.
  • At x=π4x = \frac{\pi}{4}, f(π4)=3sin(π2)=3f\left(\frac{\pi}{4}\right) = 3\sin\left(\frac{\pi}{2}\right) = 3.
  • At x=3π4x = \frac{3\pi}{4}, f(3π4)=3sin(3π2)=3f\left(\frac{3\pi}{4}\right) = 3\sin\left(\frac{3\pi}{2}\right) = -3.

Given these points, the graph will oscillate between 33 and 3-3 with a period of π\pi, completing two cycles from 00 to 2π2\pi.

Would you like me to generate a sketch of this graph, or do you have any specific questions?


Here are 5 follow-up questions based on the above discussion:

  1. How would the graph of f(x)=3sin(2x)f(x) = 3\sin(2x) change if the amplitude were doubled?
  2. What effect does changing the period have on the frequency of the sine wave?
  3. Can you determine the phase shift of f(x)=3sin(2x)f(x) = 3\sin(2x) if we add a phase constant?
  4. How would the graph of f(x)f(x) change if it were 3cos(2x)3\cos(2x) instead of 3sin(2x)3\sin(2x)?
  5. How do you find the maximum and minimum values of the function f(x)f(x)?

Tip: When sketching trigonometric functions, always identify the amplitude, period, phase shift, and vertical shift to accurately determine the graph's shape and key points.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine Function
Amplitude
Period

Formulas

Amplitude = |a|
Period = 2π / b

Theorems

Basic properties of sine function

Suitable Grade Level

Grades 10-12