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Understanding Trigonometric Functions: Domain, Range, Asymptotes, and Odd-Even Properties

This section covers essential concepts in precalculus related to trigonometric functions. Students will learn how to graph the functions f(x) = -2tan(1/3x) and f(x) = sec(2x) + 1, determining their domain, range, asymptotes, and period. The lesson highlights properties of odd and even functions, emphasizing symmetry in graphs. Key examples include the odd nature of sine and tangent functions, and the even nature of cosine. Students will also apply these properties to solve problems and evaluate trigonometric values at specific angles.

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Understanding Trigonometric Functions: Domain, Range, Asymptotes, and Odd-Even Properties

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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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