4.5 – Graphs of the other Trigonometric Functions Tangent and Cotangent

In this section, you will learn to: Sketch the graphs of tangent and cotangent functions

The graph of the tangent curve:

The graph of the cotangent curve:

Question: Is there an amplitude for a tangent or a cotangent function? Why or why not? No amplitude, since the two curves extend infinitely in both directions.

Period of Tangent and Cotangent Functions: THIS IS DIFFERENT

 Where do you think you need to set the left and right endpoints for a tangent graph below? Asymptotes of the tangent graph function:

 Where do you think you need to set the left and right endpoints for a cotangent graph below? Asymptotes of the cotangent graph function:

If a is positive, then there is no reflection about the x-axis. If a is negative, then there is a reflection about the x-axis. Reflection:

Effects of a on the tangent and cotangent graphs:

a) If a > 1, then the graph rises faster. b) If 0< a < 1, then the graph rises slower. Effects of a on the tangent and cotangent graphs:

Effects of c on the tangent and cotangent graphs:

The constant c determines the phase shift of the graph. Phase Shift = - c/k (or –c/b) a) If c is positive, then the shift is toward the left. b) If c is negative, then the shift is toward the right. Horizontal Translation or Phase Shift:

Effects of d on the tangent and cotangent graphs:

The constant d determines the vertical translation of the graph. a) If d is positive, then the vertical shift is upward. b) If d is negative, then the vertical shift is downward. Vertical Translation:

Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and sketch the graph.

a) Amplitude: b) Period: c) Horizontal Translation: d) Vertical Shift: e) Reflection:

e) Endpoints: Verify distance with the period:

The graph of the secant curve:

The graph of the cosecant curve:

a) Is there an amplitude for a secant or a cosecant function? Why or why not?

b)Period is c)The horizontal translation, vertical translation, and reflection all stay the same. d)It is best to sketch the cosecant and secant graph by first graphing the reciprocal functions of sine and cosine.

Find the amplitude, period, reflections, horizontal shift, vertical shift, endpoints and sketch the graph.

a) Amplitude: b) Period: c) Horizontal Translation: d) Vertical Shift: e) Reflection:

e) Endpoints: Verify distance with the period:

If a is positive, then there is no reflection about the x-axis. If a is negative, then there is a reflection about the x-axis. If b is positive, then there is no reflection about the y-axis. If b is negative, then there is a reflection about the y-axis. Reflection: