Chapter 4: Graphing & Inverse Functions Section 4.1 Basic Graphs
periodic function a function that repeats its values in regular intervals
Examine each of the following graphs. Does the graph represent a periodic function?
Yes
Yes
No
Yes
No
No
Yes
No
Yes
M m amplitude magnitude of change in the oscillating variable If the greatest value of f(x) is M and the least value of f(x) is m, then the amplitude of the graph of f(x) is M m
M m For trig functions, amplitude is distance to max or min value from the midline. M m
What is the amplitude of the periodic function? 1.75
What is the amplitude of the periodic function? 2
What is the amplitude of the periodic function? 1.5
What is the amplitude of the periodic function? none undefined
period smallest positive distance at which a function repeats For any function f(x), the smallest positive number p for which is called the period of f(x).
What is the period of the periodic function? 6.5
What is the period of the periodic function? 3
What is the period of the periodic function? 2
y-values to determine period…but the length along the x-axis. What is the period of the periodic function? 1 Look for repeating y-values to determine period…but the period is the length along the x-axis.
Sine Graph: f(q)= sin q q sine
Sine Graph: f(q)= sin q
Sine Graph: f(q)= sin q
Sine Graph: f(q)= sin q
Sine Graph: f(q)= sin q
Sine Graph: f(q)= sin q
Sine Graph: f(q)= sin q
Sine Graph: f(q)= sin q
Sine Graph: y = sin q Graph of one cycle Domain: Range: y-int: Amplitude: Period:
Cosine Graph: f(q)= cos q
Cosine Graph: f(q)= cos q
Cosine Graph: f(q)= cos q Graph of one cycle
Cosine Graph: y = cos q Graph of one cycle Domain: Range: y-int: Amplitude: Period:
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q
Tangent Graph: f(q)= tan q Graph of one cycle Graph of one cycle is usually to
Tangent Graph: y = tan q Graph of one cycle Domain: Amplitude: Range: y-int: Amplitude: Period: Asymptotes:
Cosecant Graph: y = csc q y = sin x To graph the cosecant, start with its reciprocal: sine.
Cosecant Graph: y = csc q Where sin Ѳ = 0, csc Ѳ is undefined. These are asymptotes.
Cosecant Graph: y = csc q Where sin Ѳ = 1, csc Ѳ = 1. Where sin Ѳ = -1, csc Ѳ = -1.
Cosecant Graph: y = csc q Where sin Ѳ = ½ (/6, 5/6), csc Ѳ = 2. Where sin Ѳ = -½ (7/6, 11/6), csc Ѳ = -2.
Cosecant Graph: y = csc q Repeat for reciprocals of other y-values.
Cosecant Graph: y = csc q Graph of one cycle Domain: Range: y-int: Amplitude: Period: Asymptotes:
To graph the secant, start with its reciprocal: cosine. Secant Graph: y = sec q y = cos x To graph the secant, start with its reciprocal: cosine.
Secant Graph: y = sec q Where cos Ѳ = 0, sec Ѳ is undefined. These are asymptotes.
Secant Graph: y = sec q Add points where y = 1 and y = -1. Sketch rest of graph using asymptotes.
Secant Graph: y = sec q Graph of one cycle Domain: Amplitude: Range: y-int: Amplitude: Period: Asymptotes:
Cotangent Graph: y = cot q To graph the cotangent, you could start with its reciprocal: tangent.
Cotangent Graph: y = cot q Where tan Ѳ = 0, cot Ѳ is undefined. These are asymptotes. Where tan Ѳ is undefined, cot Ѳ = 0.
Cotangent Graph: y = cot q Reciprocals points are graphed.
Cotangent Graph: y = cot q Graph of one cycle To graph the cotangent, it may be easier to remember the asymptotes and change to a downward slope.
Cotangent Graph: y = cot q Graph of one cycle Domain: Range: y-int: Amplitude: Period: Asymptotes:
Things to notice: The period for tan and cot is π. For all other trig functions, it is 2π. If the graph goes up or down to infinity, the amplitude is “undefined” or none. Asymptotes, and domain restrictions, are the same for tan and sec (π/2 + πk). Asymptotes, and domain restrictions, are the same for cot and csc (πk). All trig asymptotes have + πk, even when the period is 2π.
The End.