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Graphs of the Tangent, Cotangent, Cosecant and Secant Functions

Learn how to graph the tangent function on the interval [0, π] and explore its properties, transformations, and generalizations. Discover the periodicity, domain, range, and key points of the cotangent function. Utilize reciprocal identities to graph the cosecant and secant functions efficiently.

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Graphs of the Tangent, Cotangent, Cosecant and Secant Functions

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  1. Graphs of the Tangent, Cotangent, Cosecant and Secant Functions Section 5.5

  2. Graphing the Tangent Function • Periodicity: Only need to graph on interval [0, p] Why? • Plot points and graph

  3. Properties of the Tangent Function • Domain: All real numbers, except odd multiples of • Range: All real numbers • Odd function • Periodic, period p • x-intercepts: {…-2p, -p, 0, p, 2p, 3p, …} • y-intercept: 0 • Asymptotes occur at

  4. Generalizations - Graph of Tangent Function • Endpoints of the period: tan(x) =0 • Midpoint of the period: tan(x) has a vertical asymptote • ¼ of the period: tan(x) = value of A • ¾ of the period: tan(x) = value of -A • Tangent Function does not have an amplitude. Why?

  5. Transformations of the Graph of the Tangent Functions • Example. Use the generalizations of the graph of y = tan x to graph y = A tan (Bx) A =-2

  6. Graphing the Cotangent Function • Periodicity: Only need to graph on interval [0, p]

  7. Properties of the Cotangent Function • Domain: All real numbers, except multiples of p • Range: All real numbers • Odd function • Periodic, period p • Asymptotes occur at • x-intercepts: • y-intercept: none

  8. Generalizations for the graph of y=cot x • Endpoints of the period: Vertical asymptotes • Midpoint of the period: cot(x) = 0 • ¼ of the period: cot(x) = Value of A • ¾ of the period: cot(x)= Value of -A • y=cot(x) does not have an amplitude

  9. Graphing the Cosecant and Secant Functions • Use reciprocal identities • Graph of y = csc x

  10. Graphing the Cosecant and Secant Functions • Use reciprocal identities • Graph of y = sec x

  11. Graphing y = sec(x) and y = csc(x) • First graph y = cos(x) or y = sin(x) • Then use the reciprocal identities. • The secant and cosecant graphs can be called “clip-it and flip-it” graphs • Y = sec(x) and y = csc(x) do not have an amplitude.

  12. Key Points to Know! • How to graph the Tangent Function • Properties of the Tangent Function • Transformations of the Graph of the Tangent Functions • How to graph the Cotangent Function • Properties of the Cotangent Function • How to graph the Cosecant and Secant Functions using Sine and Cosine

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