Graphs of the Tangent

Precalculus 2

Agenda

• Do Now: Whattaya Remember about PEMDAS and RADicals?

• CW: Work on Portfolio! Yes, YOU NEED To be able to graph sine, cosine and tangent

• HW: UNIT Exam TUESDAY and WEDNESDAY!!

SWBAT:

• Accurately sketch a SINE, COSINE And TANGENT transformation, ID-ing amplitude, period, horizontal and vertical shifts!!

Precalculus 1 and 2

Agenda

• Do Now: Write the COSINE transformations!

• Class work – Geometry MCAS Review!

• HW: Review ALL Sine and Cosine Packets…remember our Unit Exam is THURSDAY!!!!

SWBAT:

• Review Geometry for MCAS

• Accurately sketch a SINE, COSINE And TANGENT transformation, ID-ing amplitude, period, horizontal and vertical shifts!!

The Graph of y = tan xx y

1 0 1

1.73

1.73

(0, 0)

3

3

6

6

What happens as x approaches ?

….it’s UNDEFINED!!

What happens as x approaches ?

….it’s UNDEFINED!!!

2

2

5 0 5

3

3

2

5,

2

3,

2,

2,

2

3

2

5 asymptotes

22

This is the graph for y = tan x.

22

3

20

22

32

y = - tan x

Consider the graph for y = - tan x

In this equation a, the numerical coefficient for the tangent, is equal to -1. The fact that a is negative causes the graph to “flip” or reflect about the x-axis.

y = a tan b (x - h)+k

b affects the period of the tangent graph.

For tangent graphs, the period can be determined by

.b

period

Conversely, when you already know the period of a tangent or cotangent graph, b can be determined by

.period

b

The distance between the asymptotes in this graph is…

Therefore, the period of this graph is also .

For all tangent graphs, the period is equal to the distance between any two consecutive vertical asymptotes.

22

3

20

22

32

22

3

20

22

32

y = tan x has no phase shift.

We designated the y-intercept, located at (0,0), as the key point.

It is important to be able to draw a tangent graph when you are given the corresponding equation. Consider the equation

Begin by looking at a, b, and c.

.6

3tan

xy

631

cba

.6

3tan

xy

The negative sign here means that the tangent graph reflects or “flips” about the x-axis. The graph will look like this.

1a

.6

3tan

xy

b = 3

3

b

periodUse b to calculate the period. Remember that the period is the distance

between vertical asymptotes.

.6

3tan

xy

6

c

This phase shift means the key point has shifted spaces to the right. It’s x-coordinate is . Also, notice that the key point is an x-intercept.

6

6

The distance between the x-intercept and the asymptotes on either side is , because it is half the period!!!! Caution: the distance to the asymptotes will not always be the same as the phase shift.

.6

3tan

xy

60

6

.6

3tan

xy

360

366

X-intercept

Vertical asymptote

.6

3tan

xy

Continue to add or subtract half of the period, , to determine the

labels for additional x-intercepts and vertical asymptotes.

6

263

Vertical asymptote

x-intercept

3

2

2360

3

2

2360

6323

2

http://www.analyzemath.com/Tangent/Tangent.html