Download y = tan x

November 11, 2014
Section 4.6 Graphs of Other Trigonometric Functions
Objective: Know how to sketch the graphs of trigonometric functions (by hand!).
.
all..
c
e
R
sin (-x) =
cos (-x) =
even or odd
odd sec (-x) =
even csc (-x) =
odd
even tan (-x) =
odd cot (-x) =
odd
Let's take a closer look at tangent.
Complete the table.
x
-π/2
-1.57
-1.5
-π/4
0
π/4
1.5
1.57
π/2
tan x
Asymptotes
sin x
tan x =
cos x
Where does cos x = 0?
Answers:-π/2, π/2, 3π/2,...so
vertical asymptotes: x=
y = tan x
Period: π
Domain: R, x = π/2 +nπ
Range: (-∞, ∞)
Vert. asymptote: x = π/2 +nπ
Symmetry: origin (odd)
*Tip: Use "dot mode" for the TI-83 if you do not want asymptotes displayed.
*Note: the key points are the asymptotes at the ends of the period and the
intercept in the middle.
November 11, 2014
Try these:
a) Graph y = 2 tan 2x
Graph
2π
3π
2
π
π
2
π
2
π
3π
2
2π
2π
3π
2
π
π
2
π
2
π
3π
2
2π
b) Graph y = -tan ( x )
2
Graph
Now, let's take a closer look at cotangent.
cot x = cos x
Where does sin x = 0?
sin x
y = cot x
Answers:0, π, 2π,...so
vertical asympt
Period: π
Domain: R, x = nπ
Range: (-∞, ∞)
Vert. asymptote: x = nπ
Symmetry: origin (odd)
November 11, 2014
Ex: Graph y = -2 cot 2x
Graph
2π
General equations:
y = d + a tan (bx-c)
3π
2
π
π
2
π
2
π
3π
2
2π
y = d + a cot (bx-c)
Amplitude ( a ) - not defined
If a < 0, then the graph is a reflection in the x-axis.
Vertical shift (d)
Phase shift (c/b) - horizontal shift (according to period)
Period: π/b - distance between asymptotes
*Should sketch 2 additional cycles to left and right.
Asymptotes:
Solve bx - c = -π/2
bx - c = π/2
Solve
bx - c = 0
bx - c = π
x-intercepts:
Solve sin(bx-c) = 0
Solve cos(bx-c) = 0
or find midpoint between asymptotes.
November 11, 2014
Graphs of Reciprocal Functions
1
y = csc x = sin
x
Period: 2π
Domain: R, x = nπ
Range: (-∞, 1] ∪[1, ∞)
Vert. asymptote: x = nπ
Symmetry: origin (odd)
*"hills" and "valleys"
1
y = sec x = cos
x
Period: 2π
Domain: R, x = π/2 +nπ
Range: (-∞, -1] ∪ [1, ∞)
Vert. asymptote: x = π/2 +nπ
Symmetry: y-axis (even)
To graph sec or csc functions...
Make a sketch of its reciprocal fct.
x-int. = asymptotes of sec x or csc x
max = rel. min and min. = rel. max.
November 11, 2014
Try these:
a) Graph y = 2 csc ( x )
2
Graph
4π
3π
2π
π
π
2π
3π
4π
2π
3π
2
π
π
2
π
2
π
3π
2
2π
b) Graph y = sec (x + π)
Graph
Damped Trig. Graphs
Graph the following on your calculator:
a) f(x) = x sin x
b) f(x) = (x2+1)-1 sin x
Graph a)
damping factor
Graph b)
To graph by hand...
1. Sketch y = ±(damping factor). They become the guides.
2. Sketch the graph of the trig. function "between" these guides.
3. The graph touches the guides at points that would have been
rel. max. and min-values of the trig. fct. The x-intercepts stay the
same.