PreCalculus: Lesson Plan * 9/06/2006

• Quickstart p.333 #95 (use pp. 191, 199 to help refresh your memory), 99, 101
• 4.5 Quiz - Graphing Sine and cosine functions [15 points]
  
  1. Determine the period, amplitude, and period-interval for
     y = (-1/2)sin[(3x/2) - (1/2)]
     [3 points]
     
  2. Sketch the graph of
     y = -2sin(2x + π)
     Show all work.
     [5 points]
  3. Sketch the graph of
     y = 2 + cos( x - π/4)
     Show all work.
     [5 points]
  4. Write a sine equation with a horizontal shift of π and an amplitude of 3.
     [2 points]
     
• Lesson
  (4.5) Mathematical Modeling
  
  1. Example: Find a trigonometric function to model the data in the following table
     x | 0 | π/2 | π | 3π/2 | 2π
     y | 2 | 4 | 2 | 0 | 2
     Solution: plot the 5 points. The graph starts at the intercept, so a sine function is a good choice. By examining the graph we see the amplitude is 2, the period is 2π, the graph is not shifted horizontally, and it is shifted up 2 units. So, a=2, b=1, c=0, d=2. So the equation is y = 2 + 2sinx
  2. (p.331 #75) Using a trig function to model real life data.
     Solution:
     a) amplitude is 0.85, period is 6, period-interval is [0,6], key points are (0,0), (3/2, 0.85), (3, 0), (9/2, -0.85), (6, 0). Sketch points and connect with a smooth curve.
     b) One full breath cycle is an inhale and an exhale. This corresponds to one period or 6 seconds.
     c) 60 seconds in a minute, divided by 6 seconds per cycle, that is 10 cycles per minute.
     
  (4.6) Graphs of Other Trig Functions
  
  1. tanx = sinx/cosx
     
     • undefined when cosx = 0, x = ...-π/2, π/2, 3π/2, 5π/2, ...
     • these undefined x values correspond to vertical asymptotes
     • Sketching y = a tan (bx - c)
       
       1. Find key points
          a) asymptotes
          Solve bx - c = π/2 AND bx - c = -π/2
          b) intercepts
          Midpoint between two consecutive asymptotes
          
       2. a > 0, increasing
          a < 0, decreasing
          
• Homework p.341: 9, 12, 24, 41 (checked on Monday)