Study Notes: Trigonometric Graphs (Grade 11 Mathematics)

Topic Overview

Main Concept/Theme: Trigonometric graphs represent the sine, cosine, and tangent functions and their properties.
Key Learning Objectives:
Understand the basic trigonometric functions and their graphs.
Analyze the properties of trigonometric graphs, such as amplitude, period, and phase shift.
Solve problems involving trigonometric graphs, including transformations.

Trigonometric Functions: Functions defined using the angles of triangles (sine, cosine, tangent).
Amplitude: The maximum height of a wave from its central line.
Period: The distance (or angle) over which the function completes one full cycle.
Phase Shift: The horizontal shift left or right for periodic functions.
Asymptote: A line that a graph approaches but never touches.

Sine (sin): A function that takes an angle and returns the y-coordinate of the point on the unit circle.
Cosine (cos): Returns the x-coordinate of the point on the unit circle corresponding to an angle.
Tangent (tan): The ratio of sine to cosine (tan θ = sin θ/cos θ).

Sine Graph:
Starts at (0, 0).
Range: [-1, 1].
Period: (2\pi) (360 degrees).
Amplitude: 1.
Key points: (0,0), (\left(\frac{\pi}{2}, 1\right)), ((\pi, 0)), (\left(\frac{3\pi}{2}, -1\right)), (2(\pi, 0)).
Cosine Graph:
Starts at (0, 1).
Range: [-1, 1].
Period: (2\pi) (360 degrees).
Amplitude: 1.
Key points: (0,1), (\left(\frac{\pi}{2}, 0\right)), ((\pi, -1)), (\left(\frac{3\pi}{2}, 0\right)), (2(\pi, 1)).
Tangent Graph:
Starts at (0, 0) but increases indefinitely towards vertical asymptotes.
Range: All real numbers.
Period: (\pi) (180 degrees).
Asymptotes are found at odd multiples of (\frac{\pi}{2}).

Vertical Shift: Adding/subtracting a constant to the function shifts the graph up/down.
Example: (y = sin(x) + 2) shifts the sine graph up by 2.
Horizontal Shift (Phase Shift): Modifying the input of the function shifts it left/right.
Example: (y = sin(x – \frac{\pi}{2})) shifts the graph to the right by (\frac{\pi}{2}).
Reflection: Multiplying the function by -1 reflects it across the x-axis.
Example: (y = -sin(x)).

Graphing Sine and Cosine:
Sketch the graphs for (y = 2sin(x)) and (y = cos(x) – 1). Identify their amplitude, period, and any vertical shifts.
Finding the Period and Amplitude:
For the function (y = 3cos(2x)):
- Amplitude: 3
- Period: (\frac{2\pi}{2} = \pi)
Combining Transformations:
For the function (y = 4sin(x – \pi) + 1), describe:
- Amplitude: 4
- Period: (2\pi)
- Phase Shift: (\pi) to the right
- Vertical Shift: 1 up

Trigonometric functions are periodic and have specific properties defined by amplitude, period, and phase shifts.
The sine and cosine functions oscillate between -1 and 1, while tangent extends to all real numbers.
Knowing how to graph and transform these functions is vital for understanding their applications in various fields.

Multiple Choice:
What is the amplitude of the function (y = -2sin(x))?
a) -2
b) 2
c) 1
d) 0
Open-ended:
Explain how the graph of (y = cos(x – \frac{\pi}{4})) differs from (y = cos(x)).

Geometry: Understanding angles, particularly in triangles and the unit circle.
Physics: Applications of trigonometric functions, such as waves and oscillations.
Computer Science: Use of trigonometric functions in graphics and simulations.

After completing these notes, try sketching the graphs of various trigonometric functions based on different transformations. If something isn’t clear, don’t hesitate to ask your teacher for help or discuss with classmates!