Lesson Overview:

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graphing tan, cot, sec and csc Parent function, domain, range, symmetry, x-intercept, y-intercept, and max & min.

a. Tangent Function

b. Cotangent Function

c. Secant Function

d. Cosecaant Function

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Part A: Graphing Tangent Functions 

 

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Domain is X ≠ π/2 + Kπ, K is ALL INTEGERS       

Range is ALL REAL NUMBERS

Tangent graph is symmetrical with to respect to (0,0). So, the Tangent Function is ODD

The x-intercepts are {… -2π, -π, 0, π, 2π …}

The y-intercept is (0,0)

The vertical asymptote occurs at X = π/2 + Kπ

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Part B: Graphing Cotangent Functions 

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The domain is X ≠ Kπ, K is ALL INTEGERS

The range Is ALL REAL NUMBERS

Cotangent graph is symmetrical with respect to the origin (0,0). So, the Cotangent function is ODD

The x-intercepts are π/2 + Kπ

There are no y-intercepts

The vertical asymptote occurs at X = Kπ

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Part C: Graphing Cosecant Functions 

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The domain is X ≠ Kπ, K is ALL INTEGERS

The range is [1, ∞] or [-∞, 1]

Cosecant graph is symmetrical with respect to the origin (0,0). So, the Cosecant functions is ODD

The x-intercepts are {… -2π, -π, 0, π, 2π …}

The y-intercept is (0,0)

The vertical asymptote occurs at X = Kπ

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Part D: Graphing Secant Functions 

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The domain is X ≠ π/2 + Kπ, K is ALL INTEGERS

The range is [1, ∞] or [-∞, 1]

Secant graph is symmetrical with respect to the y-axis. So, the Secant graph is EVEN

The x-intercepts {… -3π/2, -π/2, π/2, 3π/2 …}

The y-intercept is (0,1)

The vertical asymptote occurs at X = π/2 + Kπ

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The Unit Circle on the SAT

 

 

 

 

First, let's use the given information to determine the function's amplitude, mid-line, and period.

Then, we should determine whether to use a sine or a cosine function, based on the point where x=0x=0x, equals, 0.

Finally, we should determine the parameters of the function's formula by considering all the above.

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The mid-line intersection is at y=-6,so this is the mid-line.

The minimum point is 3 units below the mid-line, so the amplitude is 3

The minimum point is 2.5 units to the right of the mid-line intersection, so the period is 10. 

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Since the graph intersects its mid-line at x=0 we should use the sine function and not the cosine function.

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a=-3, d=6, b=π/5

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Answer: f(x) = -3sin (π/5 *x) -6

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Trig Functions in the Real World

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Conclusion:

In order to graph trigonometric functions you first must identify the amplitude (how far up and down the graph covers) and the period of the function which is how many units a single wave is.