Graphs of Circular Functions Presentation.pptx

At the end of the lesson, the student is
able to:
- determine the domain and range of
the different circular functions;
OBJECTIVES:
- graph the six circular functions with its
amplitude, period and phase shift.

Choose the letter of the correct answer.
Activity:
1. Choose the correct equation of the
graph:
a.) y = sin x
b.) y = cos x
c.) y = - sin x
d.) y = - cos x
Correct
Answer
b

Activity:
graph:
a.) y = sin x
b.) y = cos x
c.) y = - sin x
d.) y = - cos x
Correct
Answer
c

Activity:
graph:
a.) y = sin ½ x
b.) y = 2 sin x
c.) y = sin 2x
d.) y = sin x
Correct
Answer
c

Activity:
graph:
a.) y = sin x + 1
b.) y = cos x + 1
c.) y = 2sin x
d.) y = 2cos x
Correct
Answe
r
b

Activity:
graph:
a.) y = sin x + 1
b.) y = sin (x + Π/2)
c.) y = sin (x - Π/2)
d.) y = sin x
Correct
Answer
b

Activity:
graph:
a.) y = sin 4x
b.) y = 4sin x
c.) y = sin x + 4
d.) y = sin (x + 4)
Correct
Answer
b

Activity:
graph:
a.) y = sin ½ x
b.) y = 2sin x
c.) y = sin 2x
d.) y = sin x
Correct
Answer
a

Activity:
8. Choose the correct equation of the graph:
a.) y = 4sin x
b.) y = 3sin x + 1
c.) y = sin 3x + 1
d.) y = sin x + 4
Correct
Answer
b

Activity:
graph:
a.) y = sec x
b.) y = csc x
c.) y = -sec x
d.) y = -csc x
Correct
Answer
a

Activity:
graph:
a.) y = sec x
b.) y = 2sec x
c.) y = sec 2x
d.) y = sec x + 2
Correct
Answer
d
Bac
k

Graph of the sine function
To sketch the graph of y = sin x, first locate the key
points.
These are the maximum points, the minimum
points, and the intercepts.
x 0
sin x 0 1 0 -1 0

Graph of the sine function
Then, connect the points on the graph with a smooth
curve that extends in both direction beyond the five
points.
A single cycle is called a period.
y=sin x

Graph of the cosine function
To sketch the graph of y = cos x, first locate the key
points.
These are the maximum points, the minimum
points, and the intercepts.
x 0
cos
x
0 0
1 1
-1

Graph of the cosine function
Then, connect the points on the graph with a smooth
curve that extends in both direction beyond the five
points.
A single cycle is called a period.
y=cos x

Properties of sine and cosine function
The graph of y=sin x and y=cos x have similar
properties:
1. The domain is the set of real numbers.
2. The range is the set of y-values such that -1 ≤ y ≤ 1.
3. The maximum value is 1 and the minimum value is -
1.
4. The graph is a smooth curve.
5. Each function cycles through all the values of the
range over an x-interval of 2..
6. The cycle repeats itself indefinitely in both
directions of the x-axis.

Graph of y=a sin bx and y=a cos
bx:
Example: Sketch the graph of y=2sin4x.
X 0
Y
0
0
2
2
√𝟑 - -2
-2
0
0
0
0
1.7
3
-1.73

Graph of y=a sin bx and y=a cos bx:
Example: Sketch the graph of y=2sin4x.
|a| =
amplitud
e
- The height from the center line to the peak (or trough) of a
periodic function.
amplitude
- When |a|<1. the graphs are shrunk vertically, and when |a|>1,
the graphs are stretched vertically.
2𝜋
¿𝑏∨¿=¿¿
period - The length of one full cycle.
- If 0<|b|<1. the graphs are stretched horizontally, and
when |b|>1, the graphs are shrunk horizontally.

Steps in graphing sine and cosine functions
1. Determine the amplitude |a|, and find the period to complete one cycle
of the graph, we just need to complete the graph from 0 to .
2. Divide the interval into four equal parts, and get five division points: x1=0,
x2, x3, x4 and x5 = , where x3 is the midpoint between x1 and x5, x2 is the
midpoint between x1 and x3, and x4 is the midpoint between x3 and x5.
3. Evaluate the function at each of the five x-values identified in step 2.
The points will correspond to the highest point, lowest point, and x-
intercepts of the graph.
4. Plot the points found in step 3, and join them with a smooth curve
similar to the graph of the basic sine curve.
5. Extend the graph to the right and to the left as needed.

Graphs of y = a sin b(x-c)+d and y = a cos b(x-c)+d
The graphs of y = a sinb(x-c) and y = a cos b(x-c)
Have the same shape as y = a sinbx and y = a cos bx,
respectively, but shifted c units to the right when c > 0
and shifted |c| units to the left if c < 0. The number c
is called the phase shift of the sine or cosine graph.
The graphs of y = a sinb(x-c)+d and y = a cos b(x-c)+d
Have the same amplitude, period and phase shift as
that of y = a sinb(x-c) and y = a cos b(x-c),
respectively, but shifted d units to the upward when
d > 0 and shifted |d| units to the downward when d
< 0.

Graph of the Cosecant Function:
To graph y = csc x, use the identity
At values of x for which sin x = 0, the cosecant function is
undefined and its graph has vertical asymptotes.
Properties of y = csc x:
1. domain: all real x, x ≠ k
2. range:
3. period:
4. vertical asymptotes:
where sine is zero.

Graph of the Secant Function:
To graph y = sec x, use the identity
At values of x for which cos x = 0, the secant function is
Properties of y = sec x:
1. domain: all real x, x ≠ k
2. range:
3. period:

Graph of the Tangent Function:
To graph y = tan x, use the identity
At values of x for which cos x = 0, the tangent function is
Properties of y = tan x:
1. domain: all real x, x ≠ n
2. range:
3. period:

Graph of the Cotangent Function:
To graph y = cot x, use the identity
At values of x for which sin x = 0, the cotangent function is
Properties of y = cot x:
1. domain: all real x, x ≠ n
2. range:
3. period:
vertical

Steps in graphing tangent and cotangent
functions
1. Determine the period , then we draw one cycle of the graph on
for y = a tan bx, and on for y = a cot bx.
2. Determine the two adjacent vertical asymptotes. For y = a tan
bx, . For y = a cot bx, x = 0 and x =
4. Evaluate the function at each of these x-values identified in step 3.
The points will correspond to the signs and x-intercepts of the graph.
5. Plot the points found in step 3, and join them with a smooth curve
similar to the, graph of the basic sine curve. Extend the graph to the
right and to the left as needed.
3. Divide the interval formed by the vertical asymptotes in Step 2 into four
equal parts, and get three division points exclusively between the
asymptotes.
In general, to sketch the graphs of y = a tan
bx and y = a cot bx, a ≠ 0 and b > 0, we may
proceed with the following steps:

THANK
YOU!!
Prepared by:
GEOVANNI S. DELOS
REYES
SST III

Graphs of Circular Functions Presentation.pptx

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