Math Problem Statement
Sketch the graph of the function y=6cos(angle teta - pii/4)-3
Solution
To sketch the graph of the function , let's break down its components step by step:
1. Basic Cosine Graph:
The basic cosine function has:
- Amplitude: 1
- Period:
- No horizontal or vertical shifts.
2. Adjustments to the Basic Cosine Graph:
a. Amplitude:
In this case, the amplitude is multiplied by 6, so the graph will stretch vertically. The maximum value is and the minimum is .
b. Phase Shift:
The phase shift is to the right due to the term . This means the graph is shifted units to the right along the -axis.
c. Vertical Shift:
The function has a vertical shift of , meaning the entire graph will be shifted down by 3 units.
3. Characteristics of the Graph:
- Amplitude: 6
- Period: The period remains , as the coefficient of inside the cosine function is still 1.
- Phase Shift: to the right.
- Vertical Shift: Downward by 3 units.
4. Important Points:
- Maximum: occurs when .
- Minimum: occurs when .
- Midline: The horizontal midline is .
Sketch Instructions:
- Start by drawing the midline at .
- Mark the maximum point at and the minimum point at .
- Draw the cosine wave starting from , with its period being .
- Make sure the graph completes one cycle from to .
Would you like me to generate the graph, or do you need more details on any specific step?
Related Questions:
- How do you find the period of any trigonometric function?
- How does a vertical shift affect the graph of a trigonometric function?
- What happens to the graph if we change the amplitude of the cosine function?
- How can we determine the phase shift for any trigonometric function?
- What is the relationship between the maximum and minimum values of a cosine function and its amplitude?
Tip: To sketch a trigonometric graph effectively, always identify the amplitude, period, phase shift, and vertical shift first.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Graphing Trigonometric Functions
Formulas
y = A cos(B(θ - C)) + D
Amplitude = |A|
Period = 2π / |B|
Phase Shift = C
Vertical Shift = D
Theorems
Cosine Function Properties
Trigonometric Function Transformations
Suitable Grade Level
Grades 10-12
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