Graphing Sine and Cosine Functions - Mrs. Allison's … › uploads › 5 › 7 › 3 › 6 ›...
Transcript of Graphing Sine and Cosine Functions - Mrs. Allison's … › uploads › 5 › 7 › 3 › 6 ›...
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Accelerated Precalculus Name: _____________________________ Introducing Sine and Cosine Functions Write the value of the trig function for each angle using a decimal rounded to the nearest thousandth and plot the points. Then sketch a smooth curve and identify the characteristics using interval notation when appropriate.
x 0 6
4
3
2
23
34
56
76
54
43
32
53
74
116
2
siny x
1) sinf x x
Domain:_________________
Absolute minimum:_______
Absolute maximum:_______
Range:__________________
Zeros 2 , 2 :_________
_______________________
Symmetry where?_________
Odd function, even function,
or neither?_______________
2
x 0 6
4
3
2
23
34
56
76
54
43
32
53
74
116
2
cosy x
2) cosf x x
Domain:_________________
Absolute minimum:_______
Absolute maximum:_______
Range:__________________
Zeros 2 , 2 :_________
_______________________
Symmetry (where)?________
Odd function, even function,
or neither?_______________
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The general forms of the sine and cosine functions are sin ( ) y a b x c d and cos ( ) y a b x c d .
Make a table and graph the parent function. Using your GC, graph the new function and complete the table using zeros, max and min points. Label both axes! 3) cosg x x
x 2 3
2
2
0
2
2
3 2
cosf x x
x 2
cosg x x
Domain:_________________
Absolute minimum:_______
Absolute maximum:_______
Range:__________________
Zeros 2 , 2 :_________
_______________________
Symmetry (where)?________
Odd function, even function,
or neither?_______________
Given general form cos ( ) y a b x c d and cosg x x , complete the following: ______, ______, ______, ______a b c d .
What transformation has occurred from cosf x x (parent function) to g x ?
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4) 3sing x x
x 2 3
2
2
0
2
2
3 2
sinf x x
x 2
3sing x x
Domain:_________________
Absolute minimum:_______
Absolute maximum:_______
Range:__________________
Zeros 2 , 2 :_________
_______________________
Symmetry (where)?________
Odd function, even function,
or neither?_______________
Given general form sin ( ) y a b x c d and 3sing x x , complete the following: ______, ______, ______, ______a b c d .
What transformation has occurred from sinf x x (parent function) to g x ?
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5) sin 2g x x
x 2 3
2
2
0
2
2
3 2
sinf x x
x 2
sin 2g x x
Domain:_________________
Absolute minimum:_______
Absolute maximum:_______
Range:__________________
Zeros 2 , 2 :_________
_______________________
Symmetry (where)?________
Odd function, even function,
or neither?_______________
Given general form sin ( ) y a b x c d and sin 2g x x , complete the following: ______, ______, ______, ______a b c d .
What transformation has occurred from sinf x x to g x ?
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6) cos 2g x x
x 2 3
2
2
0
2
2
3 2
cosf x x
x
cos 2g x x
Domain:_________________
Absolute minimum:_______
Absolute maximum:_______
Range:__________________
Zeros 2 , 2 :_________
_______________________
Symmetry (where)?________
Odd function, even function,
or neither?_______________
Given general form cos ( ) y a b x c d and cos 2g x x , complete the following: ______, ______, ______, ______a b c d .
What transformation has occurred from cosf x x to g x ?
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7) sin2
g x x
x 2 3
2
2
0
2
2
3 2
sinf x x
x 3
2
sin2
g x x
Domain:_________________
Absolute minimum:_______
Absolute maximum:_______
Range:__________________
Zeros 2 , 2 :_________
_______________________
Symmetry (where)?________
Odd function, even function,
or neither?_______________
Given gen. form sin ( ) y a b x c d and sin2
g x x
, complete the following: ______, ______, ______, ______a b c d .
What transformation has occurred from sinf x x to g x ?
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Properties of Sine and Cosine Functions and Their Graphs
f(x)=sin(x) Period: 2 Odd function: f(-x) = -f(x) Symmetric: origin Domain: , Range: 1,1 Zeros: Multiples of
f(x)=cos(x) Period: 2 Even function: f(-x) = f(x) Symmetric: y-axis Domain: , Range: 1,1
Zeros: Odd Multiples of 2
General Forms:
sin ( ) y a b x c d cos ( ) y a b x c d where a, b, c, and d are constants and neither a nor b equal 0.
amplitude (height of the graph): a midline: y = d
period (how long does it take for one complete graph to form): 2
b
phase shift (right or left): c vertical shift (up or down): d a < 0 is a vertical reflection b < 0 is a horizontal reflection
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Expanding and Compressing
The constant factor a in each function • Expands the graph vertically if • Compresses the graph vertically if
Amplitude of Sine and Cosine Functions
The amplitude of a sinusoidal function is half the distance between the maximum and minimum values of the function or half the height of the wave.
midline: y = d
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The constant factor b in each function
• Compresses the graph horizontally if 1b
• Expands the graph horizontally if 1b
Period of Sine and Cosine Functions
The period of a sinusoidal function is the distance between any two sets of repeating points on the graph of the function.
y = cos 0.5x y = cos x y = cos 2x
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Phase Shift of Sine and Cosine Functions
The phase shift of a sinusoidal function is the difference between the horizontal position of the function and that of an otherwise similar sinusoidal function.
phase shift = c
Vertical Shift of Sine and Cosine Functions
The vertical shift of a sinusoidal function moves the graph up and down.
If 0d the function moves up d units.
If 0d the function moves down d units.
After a vertical shift, a new horizontal axis known as the midline becomes the reference line or equilibrium point about which the graph oscillates.
In order to graph Sine and Cosine functions using transformations, begin with the parent table of values. The new x-values are found using 1x c
b,
and the new y-values are found using ay d .
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PUT DOWN THE TECHNOLOGY! Next page: For each function, fill in the appropriate blanks and graph two complete periods. 8) 2cos 1g x x ______, ______, ______, ______a b c d
x
y
1x c
b
ay d
Amplitude:_________________
Period:____________________
Phase shift:________________
Vertical Shift:_______________
Reflection:_________________
Midline:___________________
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9) 1sin 4 3
2g x x ______, ______, ______, ______a b c d
Amplitude:_________________
Period:____________________
Phase shift:________________
Vertical Shift:_______________
Reflection:_________________
Midline:___________________
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10) cos 2 12
g x x
REWRITE! ______________________________g x ______, ______, ______, ______a b c d
Amplitude:_________________
Period:____________________
Phase shift:________________
Vertical Shift:_______________
Reflection:_________________
Midline:___________________
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Graphing Sine and Cosine Function Practice Complete the chart below. If there is no shift or reflection, write none. Do not leave any spaces blank!
Function Amplitude Period Phase Shift (L or R & how many)
Vertical Shift (U or D & how many)
Reflection (V, H, or both?)
Midline Equation
1) 2cos (3 ) 1y x
2) 9sin4 xy
3) sin (6 4 )y x
4) 5cos( 2 2 ) 3y x
5) xy sin8
6) 3cos(4 2 )y x
7) 7cos 7
5y x
8) 8sin(2 )
3y x
9) 86sin5 xy
10) 4)cos(2 xy
Name: ________________________________
Date: __________________ Block: _________
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For each function, fill in the appropriate blanks and graph (separate paper) two complete periods.
11) 2cos2
y x
amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______
12) y x sin 2 1 amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______ 13) 4sin (2 )y x amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______ 14) y x cos( ) 2 amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______ 15) 1)sin(5 xy amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______
16) cos 42
y x
amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______
17) xy sin2 amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______ 18) )cos(3 xy amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______
19) sin2
y x
amplitude_______period_______phase shift_______vertical shift_______reflection_______midline_______
20) 3cos4 xy amplitude_______period_______phase shift_______vertical shift_______reflection_______midline______
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GSE Accelerated Pre-Calculus Name __________________________ Identifying Graphs of Sine and Cosine Date ____________Block __________ Examine the graph below and determine the amplitude, period, phase shift, vertical shift and reflection of each. Then write an equation of each function. If your answer is different from mine; put BOTH answers in Desmos to check! 1.
Amplitude: ______________
Period: _________________
Phase Shift: _____________
Vertical Shift: ____________
Reflection: ______________
Function: _______________
2.
Amplitude: ______________
Period: _________________
Phase Shift: _____________
Vertical Shift: ____________
Reflection: ______________
Function: _______________
3.
Amplitude: ______________
Period: _________________
Phase Shift: _____________
Vertical Shift: ____________
Reflection: ______________
Function: _______________
4.
Amplitude: ______________
Period: _________________
Phase Shift: _____________
Vertical Shift: ____________
Reflection: ______________
Function: _______________
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5.
Amplitude: ______________
Period: _________________
Phase Shift: _____________
Vertical Shift: ____________
Reflection: ______________
Function: _______________
6.
Amplitude: ______________
Period: _________________
Phase Shift: _____________
Vertical Shift: ____________
Reflection: ______________
Function: _______________
Given the following information about each trig function, write a possible equation for each. 7. Sine Function
Amplitude =
Period =
Vertical Shift =
8. Sine Function Amplitude = Period = Phase Shift =
9. Cosine Function Amplitude = Period =
Phase Shift =
Vertical Shift = 3
10. Cosine Function Amplitude = Period = Phase Shift = Vertical Shift =