Graphing Periodic Functions Dr Shildneck Periodic Functions Functions
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Graphing Periodic Functions Dr. Shildneck
Periodic Functions › Functions that continue to repeat themselves are called periodic. › The length of one cycle (x-values) is called the PERIOD of the function. › All trigonometric functions are periodic.
Graphing the Basic Sinusoidal Curves › Know where your primary wave Begins – What is the first point of the wave? › Know How Long your primary wave is – What is the period of the wave? (How long is one cycle? ) › Know the Pattern of your primary wave – What are the maximums, minimums, and zeros (x-intercepts)?
The “Half-Way-In-Between” Method › Plot the first point of the wave › Go out the length of one cycle from that point and plot another point at the same level (y-value). › Use the Pattern of the function – Plot the point half way in between those points – Plot the point half way in between the first and middle point – Plot the point half way in between the middle and last point › Sketch the curve using those 5 points.
The Sine Curve (one period) › What is the Period? › What is the domain? › What is the range?
The Cosine Curve (one period) › What is the Period? › What is the domain? › What is the range?
Graphing the Other Trigonometric Functions
Graphing the Secant and Cosecant Functions
Trigonometric Relationships › What is the relationship between the sine and cosecant functions, and the cosine and secant functions? They are RECIPROCALS
Trigonometric Relationships › What are the important implications of these relationships? Undefin ed 1 › 1. Whenever the sine/cosine = 0, the cosecant/secant is ______ › 2. Whenever the sine/cosine = 1, the cosecant/secant = _______ › 3. Whenever the value of the sine/cosine is small (close to zero) the Large value of the cosecant/secant will be ______
Trigonometric Relationships › What are the important implications of these relationships? › Since the value of the cosecant/secant is undefined when the sine/cosine asympt is zero, the graph of the cosecant/secant will have ________ at those xotes values. › Since the value of the cosecant/secant is one when the value of the sine/cosine is one, the graphs have (only) those points in common.
Graphing the Reciprocals of Sinusoids › Since the secant and cosecant functions are the reciprocals of the cosine and sine functions, we can use those functions to guide our graphs. › First, lightly sketch the graph of the sine (for cosecant) or cosine (for secant) utilizing all of the same transformations that can by picked out from the function. › Second, sketch asymptotes for each of the “zeros” of the sinusoidal function (since the reciprocal of zero is undefined). › Finally sketch a “U” between each asymptote using the max/min of the sinusoid as the vertex. Each “U” should come in along the asymptote, level off, pass through the max/min point, then go out along the other asymptote.
› What is the Period? › What is the domain? › What is the range?
› What is the Period? › What is the domain? › What is the range?
Graphing the Tangent and Cotangent Functions
Characteristics of Tangent Functions ›
Tangent vs. Cotangent › The graphs of the tangent and cotangent functions are very similar (like the sine and cosine, like the secant and cosecant). › The differences between them are simply a vertical reflection and a horizontal shift. › By remembering a few key aspects of the graphs (their patterns), you can easily and quickly sketch the graphs of tangent and cotangent functions.
Tangent vs. Cotangent – Key Components › At x=0: - The tangent has a “zero. ” › - The cotangent has an asymptote. › - This component repeats after the period. › - Half way in between the graph has the “other” › characteristic. › The tangent goes up (left to right), the cotangent goes down. › Half way in between the asymptotes and zeros, the curve has a point one unit up (or down) from the axis. (Later, this will give us our “stretch” factor. )
› What is the Period? › What is the domain? › What is the range?
› What is the Period? › What is the domain? › What is the range?
Graphing on Given Domains › Sketch the graph of the primary period. › Follow the pattern of the 5 points and extend it to cover the given domain. › Sketch the curve › My rule is that you have all of the requested domain covered. You can always have more, but you must have at least the required set of x-values completed.
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