Transformations of Circular Functions
Precalculus HS Mathematics Unit: 04 Lesson: 02
Transformations of Circular Functions The graph of f(x) = sin x is shown on each grid (dotted). With the help of a calculator, sketch each transformation of this parent circular function. (Use ZOOM 7: ZTrig.) 1)
f(x) = 2 sin x
5)
f(x) = sin x + 1
2)
f(x) = 4 sin x
6)
f(x) = sin x + 2
3)
f(x) = 0.5 sin x
7)
f(x) = sin x – 1
4)
f(x) = -3 sin x
8)
f(x) = 2 sin x + 2
©2012, TESCCC
07/25/12
page 1 of 4
Precalculus HS Mathematics Unit: 04 Lesson: 02
Transformations of Circular Functions 9)
Explain the graphical effects of a and d on functions of the form f ( x ) a sin x d .
10)
Find functions of the form f ( x ) a sin x d to match these graphs. Check with a calculator. f(x) = ___sin x + ____
f(x) = ___sin x + ____
The graph of f(x) = cos x is shown (dotted) on each grid below. With the help of a calculator, sketch each transformation of this parent circular function. (Use a WINDOW with -3 x 6 and -2 y 2, with the x-axis in numeric radian scale.) 11) f(x) = cos (x + 1) 13) f(x) = cos (x 2)
12)
f(x) = cos (x 1)
15)
What value of c would make f(x) = cos (x c) equal to f(x) = sin x ?
©2012, TESCCC
14)
07/25/12
Explain the graphical effects of c on functions of the form f ( x ) cos( x c ) .
page 2 of 4
Precalculus HS Mathematics Unit: 04 Lesson: 02
Transformations of Circular Functions WINDOW Xmin = 0 Ymin = -2 Ymax = 2 Xmax = 4 Yscl = 1 Xscl = /2
The period of a sinusoidal function can be determined by taking the length of the x-interval shown on the graph and dividing it by the number of cycles completed in the interval. In the example to the right, the period of cosine is: (“x” distance) / (# of cycles) = 4 / 2 = 2
Length: 4
Cycle 1
Cycle 2
Use this information to complete the chart for functions of the form f ( x ) cos( bx ) . Function
Sketch
Interval
# Cycles
16)
f ( x ) cos(2 x )
4
17)
f ( x ) cos(3 x )
4
18)
f ( x ) cos( 21 x )
4
19)
f ( x ) cos( 54 x )
4
20)
Explain the graphical effects of b on functions of the form f ( x ) cos( bx ) .
©2012, TESCCC
07/25/12
Period
page 3 of 4
Precalculus HS Mathematics Unit: 04 Lesson: 02
Transformations of Circular Functions Complete these notes to summarize the effects of a, b, c, and d on functions of the form f ( x ) a sin( b( x c )) d or f ( x ) a cos( b( x c )) d .
a
d
The a-value is a ______________ that the sinusoid
The d-value indicates the vertical shift (the _______________
reaches _______________________________ its central horizontal axis (middle line).
of the sinusoid’s central horizontal axis or, middle line), at _________.
f ( x ) a sin( b( x c )) d
f ( x ) a cos( b( x c )) d
b
c
The b-value is used to determine the___________ The c-value indicates the phase shift (the _______________where the period begins),
which is the _____________________ in the x-direction required to complete ____________.
(at _______________)
Frequency is equal to the reciprocal of the period. ©2012, TESCCC
07/25/12
page 4 of 4
Transformations of Circular Functions The graph of f(x) = sin x is shown on each grid (dotted). With the help of a calculator, sketch each transformation of this parent circular function. (Use ZOOM 7: ZTrig.) 1)
f(x) = 2 sin x
5)
f(x) = sin x + 1
2)
f(x) = 4 sin x
6)
f(x) = sin x + 2
3)
f(x) = 0.5 sin x
7)
f(x) = sin x – 1
4)
f(x) = -3 sin x
8)
f(x) = 2 sin x + 2
©2012, TESCCC
07/25/12
page 1 of 4
Precalculus HS Mathematics Unit: 04 Lesson: 02
Transformations of Circular Functions 9)
Explain the graphical effects of a and d on functions of the form f ( x ) a sin x d .
10)
Find functions of the form f ( x ) a sin x d to match these graphs. Check with a calculator. f(x) = ___sin x + ____
f(x) = ___sin x + ____
The graph of f(x) = cos x is shown (dotted) on each grid below. With the help of a calculator, sketch each transformation of this parent circular function. (Use a WINDOW with -3 x 6 and -2 y 2, with the x-axis in numeric radian scale.) 11) f(x) = cos (x + 1) 13) f(x) = cos (x 2)
12)
f(x) = cos (x 1)
15)
What value of c would make f(x) = cos (x c) equal to f(x) = sin x ?
©2012, TESCCC
14)
07/25/12
Explain the graphical effects of c on functions of the form f ( x ) cos( x c ) .
page 2 of 4
Precalculus HS Mathematics Unit: 04 Lesson: 02
Transformations of Circular Functions WINDOW Xmin = 0 Ymin = -2 Ymax = 2 Xmax = 4 Yscl = 1 Xscl = /2
The period of a sinusoidal function can be determined by taking the length of the x-interval shown on the graph and dividing it by the number of cycles completed in the interval. In the example to the right, the period of cosine is: (“x” distance) / (# of cycles) = 4 / 2 = 2
Length: 4
Cycle 1
Cycle 2
Use this information to complete the chart for functions of the form f ( x ) cos( bx ) . Function
Sketch
Interval
# Cycles
16)
f ( x ) cos(2 x )
4
17)
f ( x ) cos(3 x )
4
18)
f ( x ) cos( 21 x )
4
19)
f ( x ) cos( 54 x )
4
20)
Explain the graphical effects of b on functions of the form f ( x ) cos( bx ) .
©2012, TESCCC
07/25/12
Period
page 3 of 4
Precalculus HS Mathematics Unit: 04 Lesson: 02
Transformations of Circular Functions Complete these notes to summarize the effects of a, b, c, and d on functions of the form f ( x ) a sin( b( x c )) d or f ( x ) a cos( b( x c )) d .
a
d
The a-value is a ______________ that the sinusoid
The d-value indicates the vertical shift (the _______________
reaches _______________________________ its central horizontal axis (middle line).
of the sinusoid’s central horizontal axis or, middle line), at _________.
f ( x ) a sin( b( x c )) d
f ( x ) a cos( b( x c )) d
b
c
The b-value is used to determine the___________ The c-value indicates the phase shift (the _______________where the period begins),
which is the _____________________ in the x-direction required to complete ____________.
(at _______________)
Frequency is equal to the reciprocal of the period. ©2012, TESCCC
07/25/12
page 4 of 4
