2.7 TAN, COT, SEC and CSC Graphs
2.7 TAN, COT, SEC and CSC Graphs
HOMEWORK: 2.7: 21-24, 29-32, 35, 36, 39, 40
2.8: 19, 20, 22-25
y = tan x
Key Points: (-π/4, -1), (0,0), (π/4, 1)
Key asymptotes: x = π/2 + kπ
y = cot x
Key points: (π/4, 1), (π/2, 0), (3π/4, -1)
Key Asymptotes: x = 0 + kπ
y = sec x and y = csc x
To get these graphs, we will just use the identities:
sec x = 1/ cos xand csc x = 1 / sin x
y = sec x
y= csc x
To graph or
Example: Graph y = 6 tan(2x – π/4) + 1 and clearly label all important features. State the equations of the asymptotes and the period.
Period ______Phase shift ______
Asymptotes in one cycle ______
General equation of asymptotes ______
Range ______
Example: Graph y = 3 cot(5x) and clearly label all important features. State the equations of all asymptotes and the period.
Period ______Phase shift ______
Asymptotes in one cycle ______
General equation of asymptotes ______
Range ______
Example: Graph y = 2sec(3x – π/4). Clearly label the coordinates of all key points and the asymptotes. State the equations of the asymptotes and period.
Period ______Phase shift ______
Asymptotes in one cycle ______
General equation of asymptotes ______
Range ______
Example: Graph y = -4csc(2x – π/3). Clearly label the coordinates of all key points and the asymptotes. State the equations of the asymptotes and period.
Period ______Phase shift ______
Asymptotes in one cycle ______
General equation of asymptotes ______
Range ______
Example: Graph y =1 - 3sec(5x + π/2). Clearly label the coordinates of all key points and the asymptotes. State the equations of the asymptotes and period.
Period ______Phase shift ______
Asymptotes in one cycle ______
General equation of asymptotes ______
Range ______
Example: Given y = - csc(x/2 + π/3). State the equations of the asymptotes and period in one cycle.
Period ______Phase shift ______
Asymptotes in one cycle ______
General equation of asymptotes ______
Range ______
Example: Graph y = - 5tan(3x + π/2). Clearly label the coordinates of all key points and the asymptotes. State the equations of the asymptotes and period.
Period ______Phase shift ______
Asymptotes in one cycle ______
General equation of asymptotes ______
Range ______
Example: Given y = 1-5cot(4x + π/2). State the equations of the asymptotes and period.
Period ______Phase shift ______
Asymptotes in one cycle ______
General equation of asymptotes ______
Range ______
The above graphs are of the equation
or and
Determine which equation and state if A and B are positive or negative
The above graphs are of the equation
or and & A>0. Determine which equation and if is
0 + k*period or not 0 + k*period
Strategy for graphing trig functions
Step 1: Identify A, B, and base graph
{Note: Remember minus sign in equation!}
Step 2: Find Amp., period and phase shift
(sine, cosine only)
**if base sin, cos, sec, or csc.
** if base tan or cot
p.s.
Step 3: Find base points
Sine: (0, 0);; ; ;
Cosine: (0, 1);; ; ;
Secant: Graph and draw asymptotes at zeros and cups toward asymptotes. If B, shift vertically.
Cosecant: Graph and draw asymptotes at zeros and cups toward asymptotes. If B, shift vertically
Tangent: ,, , ,
Cotangent:, ,, ,
Remember, if y value is , then there is an asymptote at x = the x coordinate!
Step 4: Transform the key points {Watch order!}
Horiz stretch: or
Horiz shift: {NOT just }
Vert stretch: {NOT amp!}
Vert shift:
Step 5: Plot resulting points and/or asymptotes and sketch graph.
Step 6: Draw additional cycle if needed. {Add new period to x values or use scale}
Step 7: Verify amp, period, phase shift and asymptotes etc. are correct for your graph.Check on your calculator.
Strategy to find asymptotes
Method 1: Transform the original asymptotes of the function.
For tangent, see where and went
For cotangent, see where and went
For secant, find the zeros of
For cosecant, find the zeros of
Forgeneral asymptotes, find the primary asymptotes in one cycle and add k* new period
Method 2:Set = to old asymp. of base function + k* period.
,
set and and solve for x.
set and and solve for x.
set and and solve for x.
set and and solve for x.
True or False. If the answer is false, correct it
T FThe period of is
T F The phase shift of is
T F The amplitude of of is 2
T F Two consecutive asymptotes of are
and
T F Two consecutive asymptotes of are
and
T F Two consecutive asymptotes of are
and
T F The general equation of the asymp. of
are and
T F The general equation of the asymptotes of
are and
T F The general equation of the asymptotes of
are and
T F If & A< 0, then the graph of
could look like
T F If & A< 0, then the graph of
could look like
T F The period of is
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