Math Problem Statement
Solution
Let's solve the problem step by step:
Given Function:
a. Amplitude of
The amplitude of a sine function is given by the absolute value of the coefficient .
For , the amplitude is:
b. Period of
The period of a sine function is given by , where is the coefficient of inside the sine function.
For , the coefficient is 2. Therefore, the period is:
c. Sketch the graph of for
To sketch the graph:
- The function has an amplitude of 3, so the graph will oscillate between and .
- The period is , so one complete cycle of the sine wave will occur over the interval . Since ranges from to , the graph will show two complete cycles.
Graph Characteristics:
- At , .
- At , .
- At , .
- At , .
Given these points, the graph will oscillate between and with a period of , completing two cycles from to .
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:
- How would the graph of change if the amplitude were doubled?
- What effect does changing the period have on the frequency of the sine wave?
- Can you determine the phase shift of if we add a phase constant?
- How would the graph of change if it were instead of ?
- How do you find the maximum and minimum values of the function ?
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