Grade 11 Functions (MCR3U) Unit 4: Trigonometric Functions Graphing Trig. functions (Amplitude and Period) Mr. Choi © 2018 E. Choi – MCR3U - All Rights Reserved

Periodic Functions = 0 A function is periodic if it has a pattern of y-values that repeats at regular intervals. (Repeated data forms a periodic function). A periodic function has a self-repeating graph. The cycle of a graph is the smallest complete repeating pattern of the graph which can begin at any point on the graph. The horizontal distance of one cycle is called the period of the function. (Period = 6 in the diagram) The horizontal line that is halfway between the maximum and the minimum values of a periodic curve is called the axis of the curve. The equation of the axis of the curve is Graphing trig functions (Amplitude and Period) © © 2018 E. Choi – MCR3U - All Rights Reserved

Periodic Functions The magnitude of the vertical distance from the axis of the curve to either maximum or minimum value is called the amplitude of the function. (Half of the vertical distance) The amplitude, a is calculated as The amplitude is always positive. (Amplitude = 2 in the diagram) A function is periodic if when p equals the period of the function. This is just a different way of saying that the y - values of the function must repeat at each horizontal distance of p from any starting point on the function. Graphing trig functions (Amplitude and Period) © © 2018 E. Choi – MCR3U - All Rights Reserved

Sine Function 1 360o 2𝜋 𝑟𝑎𝑑 1 −1 Axis of curve : Amplitude: Period: Maximum: Minimum: Domain: Range: x-intercepts: y-intercept: 1 360o 2𝜋 𝑟𝑎𝑑 1 −1 𝑛𝜋 𝑟𝑎𝑑, 𝑛∈𝑍 Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Cosine Function 1 360o 2𝜋 𝑟𝑎𝑑 1 −1 Axis of curve : Amplitude: Period: Maximum: Minimum: Domain: Range: x-intercepts: y-intercept: 1 360o 2𝜋 𝑟𝑎𝑑 1 −1 𝜋 2 +𝑛𝜋 𝑟𝑎𝑑, 𝑛∈𝑍 Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Tangent Function 𝑁𝐴 180o 𝜋 𝑟𝑎𝑑 ∞ −∞ Axis of curve : Amplitude: Period: Maximum: Minimum: Domain: Range: x-intercepts: y-intercept: 𝑁𝐴 180o 𝜋 𝑟𝑎𝑑 ∞ −∞ 𝑥≠ 𝜋 2 +𝑛𝜋 𝑟𝑎𝑑, 𝑛∈𝑍 𝑛𝜋 𝑟𝑎𝑑, 𝑛∈𝑍 Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Trigonometric Graphs y = asinkx & y = acoskx (1) Divide 360 𝑜 𝑘 𝑜𝑟 2𝜋 𝑘 into 4 equal parts, and mark them on the x-axis. Graphing trig functions (Amplitude and Period) © 2017 E. Choi – MHF4U - All Rights Reserved

Trigonometric Graphs y = asinkx & y = acoskx Graphing trig functions (Amplitude and Period) © 2017 E. Choi – MHF4U - All Rights Reserved

Example 1: Sketch one cycle of the basic graph and the transformed graph of each of the following a) Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Example 1: Sketch one cycle of the basic graph and the transformed graph of each of the following b) Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Example 1: Sketch one cycle of the basic graph and the transformed graph of each of the following c) Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Example 1: Sketch one cycle of the basic graph and the transformed graph of each of the following d) Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Example 1: Sketch one cycle of the basic graph and the transformed graph of each of the following e) Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Example 1: Sketch one cycle of the basic graph and the transformed graph of each of the following f) Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Example 1: Sketch one cycle of the basic graph and the transformed graph of each of the following g) Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Homework Text Book: P. 299 #1 Work Sheet: Graphing trig. Functions (Amplitude, Period) Check the website for updates Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

End of lesson Graphing trig functions (Amplitude and Period) © 2018 E. Choi – MCR3U - All Rights Reserved

Example 3: Related Acute Angles Express the following into related acute angle, then evaluate. Coterminal and Related Angles in Radians © 2017 E. Choi – MHF4U - All Rights Reserved

Example 3: Related Acute Angles Express the following into related acute angle, then evaluate. Coterminal and Related Angles in Radians © 2017 E. Choi – MHF4U - All Rights Reserved