6.4 & 6.5 Graphing Trigonometric Functions. The smallest number p with the above property is called the period of the function.
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1 Math Periods of trigonometric functions Definition A function y f ( t) f ( t p) f ( t) 6.4 & 6.5 Graphing Trigonometric Functions = is periodic if there is a positive number p such that + = for any t. The smallest number p with the above property is called the period of the function. Recall that the terminal point P( x, y) on the unit circle determined by the central angle θ is the same as the terminal point determined by the central angle θ +. Because sine and cosine functions are defined in terms of the coordinates of P( x, y ), their values are unchanged by the addition of any integer multiple of y 0 1 x sinθ = sin θ + k, any k Z cosθ = cos θ + k, any k Z Property The functions sine and cosine have period. Exercise #1 Find the period of cosecant and secant functions. y Property The functions tangent and cotangent have period. 0 (x,y) t 1 x (-x,-y) Exercise # Find : a) sin( ) b) cos( ) 19 c) sin 3 d) tan + 4
2 Trigonometric graphs The graph of a function helps us get a better idea of its behavior. We will sketch graphs of the sine and cosine functions and certain transformations of these functions. We will also graph the other trigonometric functions. Graphs of the sine and cosine functions Let t be the central angle measured in radians on the unit circle ( t is also equal to the length of the arc). Sine and cosine functions repeat their values in any interval of length. To sketch their graphs we first sketch the graph of one period, when 0 t. The variations of sine and cosine for t between 0 and t sint cost y y 0 1 t 0 t y y 0 1 t 0 t Definition The amplitude of the graph of y is defined as 1 A= M m where M is the greatest value of y and m is the least value of y.
3 3 Exercise #3 Graph the following functions a) y = sin x b) y = cos x. Then state the domain, range, period, amplitude, intercepts, maximum, minimum, and the type of symmetry for each function.
4 Graphs of transformations of sine and cosine 4 Vertical translations Exercise #4 a) Graph y = + cos x over one period. State the domain, range, period, amplitude, intercepts, max, min. b) Graph y = sin x 1 over one period. State the domain, range, period, amplitude, intercepts, max, min.
5 Vertical stretching and compression 5 Exercise #5 Graph each function and state the domain, range, period, amplitude, intercepts, max, min. a) y = sin x b) 1 y= sin x
6 Reflection about the x-axis 6 Exercise #6 Graph each function over one period and state the domain, range, period, amplitude, intercepts, max, min. a) y = sin x b) y = 3cos x
7 Horizontal stretch and compression 7 Exercise #7 Graph each function over one period and state the domain, range, period, amplitude, intercepts, max, min. y= sin x a) 1 b) y= sin x.
8 8 y = asin k x b In general, if y = acos k the amplitude is A= a and the period is T = ( k > 0 ) ( x b) k and phase shift is b.. Exercise #8 Find the amplitude and the period and sketch a graph for each function over one period, 1 a) y = cosx c) y= sin x e) y= 3cos x+ 4 b) y = 4cos3x d) y= sin x f) y = 4+ 4sinx g) y= sin x h) y= cos x 3
9 9 Graphs of tangent, cotangent, secant and cosecant Periodic properties The functions tangent and cotangent have period : tan θ + = tanθ cot θ + = cot θ The functions cosecant and secant have period : sec θ + = secθ csc θ + = cscθ The graph of the tangent function What is the domain of the tangent function? x tanx 0 What are the asymptotes of the graph? 6 4 What kind of symmetry does the graph of the tangent function have? 3 Since it has a period of, we need only to sketch the graph on any interval of length and then repeat the pattern to the left and to the right. We sketch the graph on the interval,.
10 10 The graph of the cotangent function What is the domain of the cotangent function? What are the asymptotes of the graph? What kind of symmetry does the graph of the cotangent function have? x cotx Since it has a period of, we need only to sketch the graph on any interval of length and then repeat the pattern to the left and to the right. We sketch the graph on the interval ( 0, ).
11 11 What is the range of the function? What are the intercepts? The graph of the cosecant function The graph of the secant function What is the domain? What is the domain? What are the asymptotes of the graph? What are the asymptotes of the graph? What kind of symmetry does the graph of the cosecant function have? What kind of symmetry does the graph of the secant function have? What is the range? What is the range?
12 Since it has a period of, we need only to sketch the graph on any interval of length and then repeat the pattern to the left and to the right. We sketch the graph on the interval ( 0, ). 1
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