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Exploring Trigonometric Functions: Graphs & Properties

Learn to graph and analyze trigonometric functions sin(x), cos(x), tan(x) based on unit circle values. Understand key properties like periods, amplitudes, and phase shifts.

joelle-gillespie
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Exploring Trigonometric Functions: Graphs & Properties

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  1. 6.4.1 – Intro to graphing the trig functions

  2. Similar to other functions, we can graph the trig functions based on values that occur on the unit circle • For this section, we will the notation: • sin(x) • cos(x) • tan(x) • Etc…

  3. Basic Properties • For the input variable x, we will use values of 0 ≤ x ≤ 2π • We will evaluate each function, just like a standard function from before • Form an ordered pair, (x, f(x)) OR (x, y)

  4. Graphing sin(x) • Before we can graph sin(x), lets actually fill in the different values that occur on the unit circle

  5. Values for sin(x)

  6. Values for sin(x), continued

  7. Graphing sin(x)

  8. Graphing cos(x) • Before we can graph cos(x), lets actually fill in the different values that occur on the unit circle • Look at any similarities

  9. Values for cos(x)

  10. Values for cos(x)

  11. Graphing cos(x)

  12. Graphing tan(x) • To graph tan(x), we have to consider the equation tan(x) = sin(x)/cos(x) • Using our two tables, let’s compile a table for tan(x)

  13. Values for tan(x)

  14. Values for tan(x)

  15. Graph for tan(x)

  16. Why are there “gaps” in the tangent function? • Where else/what ever trig functions may the “gap” reappear

  17. Combine • Let’s combine the graphs for sin(x) and cos(x)

  18. Terminology • Periodic = a function f is said to be periodic if there is a positive number p such that f(x +p) = f(x) • When values repeat • Different x values for the same y-value

  19. Periods • For sin(x), cos(x), the period is 2π • For the function f(x) = sin(bx – c) or g(x) = cos(bx – c) • Period = 2π/|b|

  20. Example. Determine the period for the function f(x) = 3sin(3x – 2) • Example. Determine the period for the function g(x) = 10cos(8x + 1)

  21. Terminology Continued • Amplitude = distance between the x-axis and the maximum value of the function • For the function f(x) = asin(x) or g(x) = acos(x), the value |a| is the amplitude

  22. Example. Determine the amplitude for the function f(x) = 10sin(2x) • Example. Determine the amplitude for the function g(x) = -14.2cos(9x)

  23. Terminology, 3 • Phase Shift = a change in the starting and stopping points for the period of a function • For the function f(x) = asin(bx – c) and g(x) = acos(bx – c); • Phase Shift = c/b

  24. Example. Find the phase shift for the function f(x) = -2cos(πx + 3π) • Example. Find the phase shift for the function f(x) = 9sin(5πx - 9π)

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