1

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?_______________

3

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 ?

4

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 ?

5

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 ?

6

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 ?

7

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 ?

8

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

9

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

10

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

11

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 .

12

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:___________________

13

9) 1sin 4 3

2g x x ______, ______, ______, ______a b c d

Amplitude:_________________

Period:____________________

Phase shift:________________

Vertical Shift:_______________

Reflection:_________________

Midline:___________________

14

10) cos 2 12

g x x

REWRITE! ______________________________g x ______, ______, ______, ______a b c d

Amplitude:_________________

Period:____________________

Phase shift:________________

Vertical Shift:_______________

Reflection:_________________

Midline:___________________

15

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: _________

16

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______

17

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 =