Transcript 6.5.1 Trigonometric Graphs

6.5.1 Trigonometric Graphs
Remember 6.3 transformations
y = ±a sin(bx - h) + k
a is a dilation
h is the horizontal shift
k is the vertical shift
a is now going to be called the amplitude
h is now called the phase shift
b is a new variable we didn’t look at
before, it will be the period, or frequency
(physics term)
The book goes by:
 y = a sin(bx + c)
 a in our book will be amplitude (common to both
math and physics)
 b will be period (math term),
frequency
(physics term)
 c will be the phase shift (math term)
 horizontal translation (math theory term)
 They also discuss Vertical Translation which we
have used before but they leave the theory out
of this section, for more information refer to 6.3
Examples:
The book does a really good job of
explaining the ideas of this section.
It includes full picture color graphs please
refer to page 448 for the graphs.
y = sin (x)
y = 2 sin (x)
Notice the difference of amplitude, it has
doubled thus the range of y – values have
doubled refer to these tables:
Table set:
 Tables:





the range of values for y
For y = sin(x)
R = { y| -1 ≤ y ≤ 1}
For y = 2sin(x)
R = { y| -2 ≤ y ≤ 2}
Period:
For period it is a little tougher to imagine
On the unit circle, it would be like
increasing the rotation of a wheel
For y = sin (x)
the b value is 1 and the period is 2
The period is the point when the graph
repeats y values
2
1 period
The period is found by a linear equation
 Since, b = 1 and
 b period = 1 period = 2
 Then the period for a translation is
2
period 
b
Finding the period
 for b values -1 < b < 1
2
 we see an expansion of period ie 2 
b
1 Period (4)
 for b values 1 < b and -1 < b
2
 2
 we see a compression of period ie
b
1 period ()
Phase Shift
 Notice that we have already talked
about horizontal translations
 This is simply new terminology
Horizontal Translation right 
Horizontal Translation left 


cosx   sin  x  
2

To put it all together:
y = a sin (bx + c)
The amplitude is: |a|
2
The period is:
b
c
And the phase shift is:
b
c
Why
?
b
Notice amplitiude relates to y values since
bx + c is the argument they both relate to x
Thus we are doing a two part translation
for x values
Up till now we have only looked at b = 1 so
the Horizontal Translation was always = -c
But now that b may or may not be 1 we
must use the new formula
Homework
p 458 1, 2, 5-31 odd, 41, 61, 62