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Precalculus

Precalculus. Section 7.5. Warmup. Graph the function. State the Domain, Range, Asymptotes, and Period f(x) = -2 tan(1/3 x) f (x) = sec(2x) + 1. Warmup Answers. f(x) = -2 tan(1/3 x) Domain: Range: Asymptotes: Period:. Warmup Answers. f(x) = sec(2x) + 1 Domain: Range: Asymptotes:

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Precalculus

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  1. Precalculus Section 7.5

  2. Warmup Graph the function. State the Domain, Range, Asymptotes, and Period • f(x) = -2 tan(1/3 x) • f(x) = sec(2x) + 1

  3. Warmup Answers • f(x) = -2 tan(1/3 x) Domain: Range: Asymptotes: Period:

  4. Warmup Answers • f(x) = sec(2x) + 1 Domain: Range: Asymptotes: Period:

  5. 7.5 Lesson – Unit Circle and Properties of Trig Functions • You do not need to write down the information on this slide in your notes • We have actually already done most of this section: we have talked about the unit circle and we discussed domain, range, and period while graphing the trig functions • Today we will be adding one property: odd-even properties

  6. 7.5 Lesson – Unit Circle and Properties of Trig Functions • You do not need to attempt to copy the following graphs • Look for SYMMETRY in the graphs • Could the function be reflected over a line or a point? • Example: Reflected over y-axis or reflected over origin

  7. What is the graph of f(x) = x ?

  8. What is the graph of f(x) = x2 ?

  9. What is the graph of f(x) = x3 ?

  10. What is the graph of f(x) = x4 ?

  11. What is the graph of f(x) = x5 ?

  12. What is the graph of f(x) = x6 ?

  13. Do you see the pattern? • Odd powers: • Even powers:

  14. (Write this down) • An “odd” function reflects over the origin • An “even” function reflects over the y-axis

  15. Function Notation Definitions of odd and even functions • Odd function:f(-x) = - f(x) • Even function:f(-x) = + f(x)

  16. Graphical Example (odd)

  17. Graphical Example (even)

  18. Is sine odd or even? • Graph the base graph of sine and determine if it is odd or even

  19. Is sine odd or even?

  20. Is sine odd or even? • Sine is odd • On your green sheet of trig rules, find the odd-even properties and write:sin(- x) = - sin(x)

  21. Is cosine odd or even? • Graph the base graph of cosine and determine if it is odd or even

  22. Is cosine odd or even?

  23. Is cosine odd or even? • Cosine is even • On the green sheet write:cos(- x) = cos(x)

  24. Is tangent odd or even? • Graph the base graph of tangent and determine if it is odd or even

  25. Is tangent odd or even?

  26. Is tangent odd or even? • Tangent is odd • On your green sheet write:tan(- x) = - tan(x)

  27. What about the other three? • The other three functions (secant, cosecant, and cotangent) will have the same property as its reciprocal • On your green sheet add the red part:sin(- x) = - sin(x) (and csc)cos(- x) = cos(x) (and sec)tan(- x) = - tan(x) (and cot)

  28. Using the odd-even properties • Find the exact value of sin(-45°)

  29. Using the odd-even properties • Find the exact value of cos(-120°)

  30. Using the odd-even properties • Find the exact value of

  31. Using the odd-even properties(try this on your own) • Find the exact value of

  32. Another Example • If f(x) = cos(x) and f(a) = ¼, find the exact value of f(-a) • Answer: • Since cosine is even, f(-a) = f(a)Since f(a) = ¼, f(-a) = ¼

  33. Example continued • If f(x) = cos(x) and f(a) = ¼, find the exact value of f(a) + f(a + 2π) • Answer: • Since the period of cosine is 2π, f(a + 2π) will equal f(a) (think about the graph and how cos(0) = cos(2π) ) • So we have f(a) + f(a + 2π) = ¼ + ¼ = 2/4 = ½

  34. HW Time • You may use the rest of the period to work on homework

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