The sketching of the sine and
cosine
functions of the form

Period = 2?/|k|

Horizontal Shift (translation) = d , to the left if (- d) is positive and to the right if (- d) is negative.

Vertical Shift (translation) = c , up if c is positive and down if c is negative.

to the rotation of ?/2 (or 90°) corresponds the point (0,1) = (cos(?/2),sin(?/2))

to the rotation of ? (or 180°) corresponds the point (-1,0) = (cos(?),sin(?))

to the rotation of 3?/2 (or 270°) corresponds the point (0,-1) = (cos(3?/2), sin(3?/2))

to the rotation of 2 ? (or 360°) corresponds the point (1,0) = (cos(2?),sin(2?)) as shown below .

Amplitude = |2| = 2

Period = 2?

Vertical Shift (translation) = 1 , up 1 unit.

Horizontal Shift (translation) = 0

We start by skeching y = cos(x) using the values of x and y from the unit circle (blue graph below).

x = 0 | ?/2 | ? | 3?/2 | 2? |

y = 1 | 0 | -1 | 0 | 1 |

We then sketch y = 2 cos(x) streching y = cos(x) by 2 (green graph below) and finally y = 2 cos(x) + 1 by shifting up 1 unit (red graph below).

Amplitude = |-2| = 2

Period = 2?

Vertical Shift (translation) = - 1 , down 1 unit.

Horizontal Shift (translation) = 0

We start by skeching y = sin(x) using the values of x and y from the unit circle (blue graph below).

x = 0 | ?/2 | ? | 3?/2 | 2? |

y = 0 | 1 | 0 | - 1 | 0 |

We then sketch y = - 2 sin(x) streching y = sin(x) by 2 and reflecting it on the x-axis(green graph below) and finally y = - 2 sin(x) - 1 by shifting down 1 unit (red graph below).

Amplitude = |3| = 3

Period = 2?/2 = ?

Vertical Shift (translation) = - 1 , down 1 unit.

Horizontal Shift: Because of the term ?/3, the graph is shifted horizontally. We first rewrite the given function as: y = 3 cos [ 2( x + ?/6)] - 1 and we can now write the shift as being equal to ?/6 to the left.

We start by skeching 3 cos(2 x) with minimum and maximum values - 3 and + 3 over one period = ? (blue graph below).

We then sketch y = 3 cos(2 x) - 1 translating the previous graph 1 unit down (green graph below). We now shift the previous graph ?/6 to the left (red graph below) so that the sketched period starts at - ?/6 and ends at - ?/6 + ? = 5?/6 which is one period = ?.

Amplitude = |- 0.2| = 0.2

Period = 2?/0.5 = 4?

Vertical Shift (translation) = 0.1 , up 0.1 unit.

Horizontal Shift: Because of the term - ?/6, the graph is shifted horizontally. We first rewrite the given function as: y = - 0.2 cos [ 0.5( x - ?/3)] + 0.1 and we can now write the shift as being equal to ?/3 to the right.

We start by skeching - 0.2 sin(0.5 x) with minimum and maximum values - 0.2 and + - 0.2 over one period = 4 ? (blue graph below).

We then sketch y = - 0.2 sin(0.5 x) + 0.1 translating the previous graph 0.1 unit up (green graph below). We then shift the previous graph ?/3 to the right (red graph below) so that the sketched period starts at ?/3 and ends at ?/3 + 4? which is one period = 4?.

Amplitude = |2| = 2

Vertical Shift (translation) = - 2 , down 2 units.

Period = 360/2 = 180°

Horizontal Shift: Because of the term - 60°, the graph is shifted horizontally. We first rewrite the given function as:y = 2 cos[2( x - 30°)] - 2 and we can now write the shift as being equal to 30° to the right.

We start by skeching y = 2 cos(2 x) with minimum and maximum values - 2 and + 2 over one period = 180° (blue graph below).

We then sketch y = 2 cos(2 x) - 2 translating the previous graph 2 units down (green graph below). We then shift the previous graph 30° to the right (red graph below) so that the sketched period starts at 30° and ends at 30° + 180° = 210° which is one period = 180°.

Amplitude = |- 2| = 2

Period = 2?/(1/3) = 6?

Vertical Shift (translation) = - 1 , down 1 unit.

Horizontal Shift: Because of the term ?/3, the graph is shifted horizontally. We first rewrite the given function as: y = - 2 sin[(1/3)(x + ?)] - 1 and we can now write the shift as being equal to ? to the left.

We start by skeching - 2 sin(x/3) with minimum and maximum values - 2 and + 2 over one period = 6 ? (blue graph below).

We then sketch y = - 2 sin(x/3) - 1 translating the previous graph 1 unit down (green graph below). We then shift the previous graph ? to the left (red graph below) so that the sketched period starts at -? and ends at 5? which is one period = 6?.

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