4.2 Translations of the Graphs of the Sine and Cosine Function Combinations of Translations Further Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = c + a sin[b(x – d)] or y = c + a cos [b(x - d)], with b>0, follow these steps. Method 1 2т Step 1 Find an interval whose length is one period " by solving the three- part inequality 0 < b(x – d) < 27. Step 2 Divide the interval into four equal parts to obtain five key x-values. Step 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and points that intersect the line y = c (“middle" points of the wave). Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|. Step 5 Draw the graph over additional periods, as needed. Method 2 Step 1 Graph y = a sin bx or y = a cos bx. The amplitude of the function is 2т lal, and the period is . Step 2 Use translations to graph the desired function. The vertical transla- tion is c units up if c > 0 and c units down if c<0. The horizontal translation (phase shift) is d units to the right if d> 0 and d units to the left if d<0. EXAMPLE 5 Graphing y = c + a sin[b(x -d] Graph y = -1 + 2 sin(4x + ) over two periods. SOLUTION We use Method 1. We must first write the expression on the right side of the equation in the form c + a sin [b(x – d)]. Rewrite by y = -1+ 2 sin(4x + 7), or y = -1+ 2 sin 4 x + factoring out 4. Step 1 Find an interval whose length is one period. Subtract 0<4x+ < 2T Three-part inequality 4 TT TT TT7 TT How do you get 4 TT Divide each part by 4. IT Subtract from each part. 4 х 4 TT Step 2 Divide the interval -, into four equal parts to obtain these x-values. 4 TT 0, Key x-values 4. 4. 8' 8. VI VI
4.2 Translations of the Graphs of the Sine and Cosine Function Combinations of Translations Further Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = c + a sin[b(x – d)] or y = c + a cos [b(x - d)], with b>0, follow these steps. Method 1 2т Step 1 Find an interval whose length is one period " by solving the three- part inequality 0 < b(x – d) < 27. Step 2 Divide the interval into four equal parts to obtain five key x-values. Step 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and points that intersect the line y = c (“middle" points of the wave). Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude |a|. Step 5 Draw the graph over additional periods, as needed. Method 2 Step 1 Graph y = a sin bx or y = a cos bx. The amplitude of the function is 2т lal, and the period is . Step 2 Use translations to graph the desired function. The vertical transla- tion is c units up if c > 0 and c units down if c<0. The horizontal translation (phase shift) is d units to the right if d> 0 and d units to the left if d<0. EXAMPLE 5 Graphing y = c + a sin[b(x -d] Graph y = -1 + 2 sin(4x + ) over two periods. SOLUTION We use Method 1. We must first write the expression on the right side of the equation in the form c + a sin [b(x – d)]. Rewrite by y = -1+ 2 sin(4x + 7), or y = -1+ 2 sin 4 x + factoring out 4. Step 1 Find an interval whose length is one period. Subtract 0<4x+ < 2T Three-part inequality 4 TT TT TT7 TT How do you get 4 TT Divide each part by 4. IT Subtract from each part. 4 х 4 TT Step 2 Divide the interval -, into four equal parts to obtain these x-values. 4 TT 0, Key x-values 4. 4. 8' 8. VI VI
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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I dont understand how you get π/2 - π/4= π/4
i attached a picture of the problem from the book
![4.2 Translations of the Graphs of the Sine and Cosine Function
Combinations of Translations
Further Guidelines for Sketching Graphs of Sine and
Cosine Functions
To graph y = c + a sin[b(x – d)] or y = c + a cos [b(x - d)], with b>0,
follow these steps.
Method 1
2т
Step 1 Find an interval whose length is one period " by solving the three-
part inequality 0 < b(x – d) < 27.
Step 2 Divide the interval into four equal parts to obtain five key x-values.
Step 3 Evaluate the function for each of the five x-values resulting from
Step 2. The points will be maximum points, minimum points, and
points that intersect the line y = c (“middle" points of the wave).
Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve
having amplitude |a|.
Step 5 Draw the graph over additional periods, as needed.
Method 2
Step 1 Graph y = a sin bx or y = a cos bx. The amplitude of the function is
2т
lal, and the period is .
Step 2 Use translations to graph the desired function. The vertical transla-
tion is c units up if c > 0 and c units down if c<0. The horizontal
translation (phase shift) is d units to the right if d> 0 and d units
to the left if d<0.
EXAMPLE 5 Graphing y = c + a sin[b(x -d]
Graph y = -1 + 2 sin(4x + ) over two periods.
SOLUTION We use Method 1. We must first write the expression on the right
side of the equation in the form c + a sin [b(x – d)].
Rewrite by
y = -1+ 2 sin(4x + 7), or y = -1+ 2 sin 4 x +
factoring out 4.
Step 1 Find an interval whose length is one period.
Subtract
0<4x+
< 2T Three-part inequality
4
TT
TT TT7
TT
How do you get
4
TT
Divide each part by 4.
IT
Subtract from each part.
4
х
4
TT
Step 2 Divide the interval -, into four equal parts to obtain these x-values.
4
TT
0,
Key x-values
4.
4.
8'
8.
VI
VI](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc82c3f0d-764b-4720-a79d-765547b2b481%2F4876a674-db3e-4366-86fd-55b1413a95e2%2F041qgdq.jpeg&w=3840&q=75)
Transcribed Image Text:4.2 Translations of the Graphs of the Sine and Cosine Function
Combinations of Translations
Further Guidelines for Sketching Graphs of Sine and
Cosine Functions
To graph y = c + a sin[b(x – d)] or y = c + a cos [b(x - d)], with b>0,
follow these steps.
Method 1
2т
Step 1 Find an interval whose length is one period " by solving the three-
part inequality 0 < b(x – d) < 27.
Step 2 Divide the interval into four equal parts to obtain five key x-values.
Step 3 Evaluate the function for each of the five x-values resulting from
Step 2. The points will be maximum points, minimum points, and
points that intersect the line y = c (“middle" points of the wave).
Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve
having amplitude |a|.
Step 5 Draw the graph over additional periods, as needed.
Method 2
Step 1 Graph y = a sin bx or y = a cos bx. The amplitude of the function is
2т
lal, and the period is .
Step 2 Use translations to graph the desired function. The vertical transla-
tion is c units up if c > 0 and c units down if c<0. The horizontal
translation (phase shift) is d units to the right if d> 0 and d units
to the left if d<0.
EXAMPLE 5 Graphing y = c + a sin[b(x -d]
Graph y = -1 + 2 sin(4x + ) over two periods.
SOLUTION We use Method 1. We must first write the expression on the right
side of the equation in the form c + a sin [b(x – d)].
Rewrite by
y = -1+ 2 sin(4x + 7), or y = -1+ 2 sin 4 x +
factoring out 4.
Step 1 Find an interval whose length is one period.
Subtract
0<4x+
< 2T Three-part inequality
4
TT
TT TT7
TT
How do you get
4
TT
Divide each part by 4.
IT
Subtract from each part.
4
х
4
TT
Step 2 Divide the interval -, into four equal parts to obtain these x-values.
4
TT
0,
Key x-values
4.
4.
8'
8.
VI
VI
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