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Understanding Amplitude, Midline, and Period in Sinusoidal Functions

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This guide explores the key concepts of sinusoidal functions, focusing on amplitude, midline, and period as described in the fourth edition of "Functions Modeling Change: A Preparation for Calculus." It explains how to determine the amplitude and midline from equations like y = A sin(t) + k and provides examples including y = 3 sin(t) + 5 and y = -0.15 cos(t) + 0.2. It also covers transformations such as horizontal shifts and helps understand how period influences the waveform. Ideal for calculus preparation.

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Understanding Amplitude, Midline, and Period in Sinusoidal Functions

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  1. 8.2 SINUSOIDAL FUNCTIONS AND THEIR GRAPHS Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  2. Amplitude and Midline The functions y = A sin t + k and y = A cost + k have amplitude |A| and the midline is the horizontal line y = k. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  3. Amplitude and Midline Example 1 State the midline and amplitude of the following sinusoidal functions: (a) y = 3 sin t + 5 (b) y = (4 − 3 cost)/20. Solution Rewrite (b) as y= 4/20 − 3/20cost y = -0.15 cost + 0.2 (a) (b) Amplitude = 3 Midline y = 5 Amplitude = 0.15 Midline y = 0.2 Graph of y = 3 sin t + 5 Graph of y = 0.2 − 0.15 cost Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  4. Period The functions y = sin(Bt) and y =cos(Bt) have period P = 2π/|B|. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  5. Finding Formulas for Functions Example 3 Find possible formulas for the functions f and g in the graphs Solution y = g(t) y = f(t) This function has period 4π This function has period 20 The graph of f resembles the graph of y = sin t except that its period is P = 4π. Using P = 2π/B Gives 4π = 2π/B so B = ½ and f(t) = sin(½ t) The graph of g resembles the graph of y = sin t except that its period is P = 20. Using P = 2π/B gives 20 = 2π/B so B = π/10 and g(t) = sin(π/10 t) Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  6. Horizontal Shift The graphs of y = sin(B(t − h)) and y = cos(B(t − h)) are the graphs of y = sin(Bt) and y = cos(Bt) shifted horizontally by h units. Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  7. Horizontal Shift Example 7 Describe in words the graph of the function g(t) = cos (3t −π/4). Solution We want the form cos(B(t − h)). Factor out a 3 to get g(t) = cos (3(t − π/12)). The period of g is 2π/3 and its graph is the graph of f = cos 3t shifted π/12 units to the right, as shown Period = 2π/3 Horizontal shift = π/12 g(t) = cos (3(t − π/12)) f(t) = cos (3t ) Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

  8. Summary of Transformations For the sinusoidal functions y = A sin(B(t − h)) + k and y = A cos(B(t − h)) + k, • |A| is the amplitude • 2π/|B| is the period • h is the horizontal shift • y = k is the midline Functions Modeling Change: A Preparation for Calculus, 4th Edition, 2011, Connally

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