Graphs of Sine and Cosine Functions
(Section 4-5)
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Section 4.5, Figure 4.42,
Graph of Sine Curve
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Section 4.5, Figure 4.43,
Graph of Cosine Curve
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Period- cycle length of a curve
5 Key points for Sine in one period
•Intercepts (3)
•Maximum (1)
•Minimum (1)
5 Key points for Cosine in one period
•Intercepts (2)
•Maximum (1)
•Minimum (2)
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Key Points in the Sine and Cosine Curves
sin x
cos x
x – intercepts
y –intercepts
Increasing intervals
Decreasing intervals
Relative Maximum
Relative Minimum
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
Range
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Amplitude – the height of a curve
a is a scaling factor – vertical stretch or vertical shrink
Range is between –a and a
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Find the period and amplitude.
Example 1 y = 2 sin 4x
-2π
2π
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Find the period and amplitude.
Example 2
-6π
y  3 cos
x
3
6π
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Find the period and amplitude.
Example 3
1
x
y  sin
4
2
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Find the period and amplitude.
Example 4
1
y  cos 6x
3
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Vertical shift
Horizontal shift
Negative on a – reflection in x-axis
Negative on b – reflection in y-axis
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Describe the relationship between the graphs of f and g. Consider the amplitudes,
periods and shifts.
Example 5 f(x) = sin x
g(x) = sin (x )+ π
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Describe the relationship between the graphs of f and g. Consider the amplitudes,
periods and shifts.
Example 6 f(x) = sin 3x
g(x) = sin (-3x)
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Describe the relationship between the graphs of f and g. Consider the amplitudes,
periods and shifts.
Example 7 f(x) = cos x
g(x) = - 3cos x
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Describe the relationship between the graphs of f and g. Consider the amplitudes,
periods and shifts.
Example 8 f(x) = cos x
g(x) = 1 cos x
2
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Describe the relationship between the graphs of f and g. Consider the amplitudes,
periods and shifts.
Example 9 f(x) = cos 2x
g(x) = 4 + cos 2x
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Describe the relationship between the graphs of f and g. Consider the amplitudes,
periods and shifts.
Example 10
g
f
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Find a and d for the function f(h) = a cos x + d such that the graph of f matches the figure.
Example 16
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Find a and d for the function f(h) = a cos x + d such that the graph of f matches the figure.
Example 17
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
HW#21/22 pg 304 (1-14all, 17-24all, 26 all, 61-64all)
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Sketch the graph of the function by hand. (Include two full periods.)
Example 11
y  2 sin x
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Sketch the graph of the function by hand. (Include two full periods.)
Example 12
y  sin
x
2
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Sketch the graph of the function by hand. (Include two full periods.)
Example 13
1
y  sin  x   2
2
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Sketch the graph of the function by hand. (Include two full periods.)
Example 14
y  3 cos2x 
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Sketch the graph of the function by hand. (Include two full periods.)
Example 15
y  2  3 cos 2 x
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
HW #23/24 Graphing Sine and Cosine WS
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Find a, b and c for the function f(h) = a sin (bx) such that the graph of f matches the figure.
Example 18 SKIP
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
Find a, b and c for the function f(h) = a sin (bx) such that the graph of f matches the figure.
Example 19 SKIP
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.
HW #37 pg 305 (27 – 33 odd, 39 – 46 all, 61-68 all, 83 – 86 all)
Essential Question: What are the period and amplitude of the sine/cosine function? How do you find them?
How do you graph sine and cos?
Students will write a summary discussing the period and amplitude of a sin or cos function and how to graph them.