11F - Graphing sine and cosine
- Both sine and cosine are wave, many-to-one, functions with periodic behavior.
- Periodic functions are functions where the same values are repeated at regular intervals: \(f(x)=f(x+P)\).
Period and amplitude:
Consider the equations \(y=A sin(nx)\) and \(y=A cos(nx)\).
Consider the equations \(y=A sin(nx)\) and \(y=A cos(nx)\).
- The amplitude is the maximum displacement from the mean value of the function. The amplitude is given by \(A\).
- The period is the length of the interval for one cycle of the function. The period is given by \(\frac{2\pi}{n}\).
11F - Example 1: Finding the period and amplitude of sine and cosine functions
State the period and amplitude for each of the following circular functions:
(a) \(y=3 sin(2x)\) (b) \(y=-4 cos(\frac{x}{2})\) (c) \(y=\frac{3}{5} sin(\frac{\pi x}{12})\) |
11F - Example 1: Video solution
11F - Example 1: Practice
Question 1: ABC Question 2: ABC 11F - Example 1: Solutions
Question 1: ABC Question 2: ABC |
Sketching graphs with dilations and reflections
Consider again the equations \(y=A sin(nx)\) and \(y=A cos(nx)\).
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■ When sketching graphs, all key features should be labelled with coordinates in exact from unless otherwise stated. |
11F - Example 2: Sketching circular functions with dilations and reflections
Sketch the graph of \(y=-4cos(\frac{x}{2})\). Clearly label all axial intercepts, maximums and minimums with their coordinates.
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11F - Example 2: Video solution
11F - Example 2: Practice
Question 1: ABC Question 2: ABC 11F - Example 2: Solutions
Question 1: ABC Question 2: ABC |
11F - Example 3: Sketching circular functions with dilations and reflections
Sketch the graph of \(y=\frac{3}{5} sin(\frac{\pi x}{12})\) for \(x \in [0,24]\). Clearly label all axial intercepts, maximums and minimums with their coordinates.
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11F - Example 3: Video solution
11F - Example 3: Practice
Question 1: ABC Question 2: ABC 11F - Example 3: Solutions
Question 1: ABC Question 2: ABC |
Sketching graphs with translations:
Consider the equations \(y=A sin(n(x-b))+c\) and \(y=A cos(n(x-b))+c\).
Consider the equations \(y=A sin(n(x-b))+c\) and \(y=A cos(n(x-b))+c\).
11F - Example 4: Sketching circular functions with vertical translations
Sketch the graph of \(y=2sin(x)-\sqrt{3}\) for \(x \in [0,4\pi]\). Clearly label all axial intercepts with their coordinates.
■ Please note that the \(x\)-intercepts of this graph were in fact found in 11D - Example 1. |
11F - Example 4: Video solution
11F - Example 4: Practice
Question 1: ABC Question 2: ABC 11F - Example 4: Solutions
Question 1: ABC Question 2: ABC |
11F - Example 5: Sketching circular functions with horizontal translations
Sketch the graph of \(y=2cos(x+\frac{\pi}{6})+1\) for \(x \in [-2\pi,\pi]\). Clearly label all axial intercepts with their coordinates.
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11F - Example 5: Video solution
11F - Example 5: Practice
Question 1: ABC Question 2: ABC 11F - Example 5: Solutions
Question 1: ABC Question 2: ABC |
Determining the rule of a wave function:
- By examining the period, amplitude and translations it is possible to determine the equation of a wave function.
- It is possible to give equivalent expressions for a wave functions in term of sine and cosine.
11F - Example 6: Determining the rule of a wave function
Consider the wave function shown below.
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11F - Example 6: Video solution
11F - Example 6: Practice
Question 1: ABC Question 2: ABC 11F - Example 6: Solutions
Question 1: ABC Question 2: ABC |