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Trigonometric Graphs Click to continue.

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You are already familiar with the basic graph of y = sin x o. There are some important points to remember. 360 o The curve has a period of It has a maximum value of 1 at 90 o. 1 90 o It has a minimum value of –1 at 270 o. 270 o It passes through the origin. O It crosses the x-axis at 180 o Click to continue. y = sin x o x y

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Let us compare the graph of y = sin x o to the family of graphs of the form y = a sin bx o + c where a, b and c are constants. We will begin by looking at graphs of the form y = a sin x o. Click to continue. For example: y = 2 sin x o, y = 3.7 sin x o or y = ½ sin x o.

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Click to continue. y = sin x o O 180 o 360 o 1 -3 -2 3 2 Here is the graph of y = sin x o. Click once to see the graph of y = 2 sin x o. y = 2 sin x o Notice the following points on the curve. It passes through the origin. It has a maximum of 2 (twice that of the normal graph). It has a minimum of –2. It has a period of 360 o. x y

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Click to continue. O 180 o 360 o 1 -3 -2 3 2 Here is the graph of y = sin x o. Click once to see the graph of y = -3 sin x o. y = -3 sin x o Notice the following points on the upside- down curve. It passes through the origin. It has a minimum of -3 (negative three times that of the normal graph). It has a maximum of 3. It has a period of 360 o. x y y = sin x o

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Click to continue. O 180 o 360 o 1 -3 -2 3 2 Here is the graph of y = sin x o. Click once to see the graph of y = 2½ sin x o. y = 2½ sin x o Notice the following points on the curve. It passes through the origin. It has a maximum of 2½ (two and a half times that of the normal graph). It has a minimum of –2½. It has a period of 360 o. x y y = sin x o

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Click to continue. O 180 o 360 o 1 -3 -2 3 2 Here is the graph of y = sin x o. Click once to see the graph of y = ½ sin x o. y = ½ sin x o Notice the following points on the curve. It passes through the origin. It has a maximum of ½ (half of the normal graph). It has a minimum of –½. It has a period of 360 o. x y y = sin x o

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Click to continue. O 180 o 360 o 1 -a a Here is the graph of y = sin x o. Click once to see the graph of y = a sin x o. y = a sin x o Notice the following points on the curve. It passes through the origin. It has a maximum of a (a times that of the normal graph). It has a minimum of –a. It has a period of 360 o. x y y = sin x o It still passes through the origin The period is unaffected. The height is now “a”.

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y = sin 15x o O 180 o 360 o 5 -15 -10 15 10 -5 x y This is the graph of which function? y = 15 sin x o y = sin x o + 15

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Which of these diagrams shows the graph of y = 7 sin x o ? x y O 180 o 360 o 7 -7 14 -14 x y O 360 o 720 o 7 -7 x y O 360 o 720 o 7 -7 x y O 180 o 360 o -3.5 3.5 540 o 180 o -7 -3.5 3.5 7

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Click to continue. For y = a sin x o only the height is affected. The graph will now have an altitude of 1 a. This is also true for y = a cos x o and y = a tan x o. 90 o 180 o 270 o 360 o O x y 1 y = cos x o 90 o 180 o 270 o 360 o O x y 1 45 o y = tan x o Here are the graphs of y = cos x o and y = tan x o.

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Click to continue. Here are some examples of the graphs of y = a cos x o. 90 o 180 o 270 o O Click for y = 2 cos x o Click for y = ¾ cos x o Click for y = - cos x o y = cos x o 360 o x y 1

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90 o 180 o 270 o 360 o O x y 1 45 o Click to continue. Here are some examples of the graphs of y = a tan x o. 450 o y = tan x o Click for y = 2tan x o 3 2 4 -2 -3 -4 Click for y = -3tan x o Notice this point

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We will now look at graphs of the form y = sin bx o. Click to continue. For example: y = sin 2x o, y = sin 3x o or y = sin ½x o.

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You are already familiar with the basic graph of y = sin x o. There are some important points to remember. 360 o 1 90 o 270 o O 180 o Click to continue. y = sin x o x y

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Click to continue. y = sin x o O 180 o 360 o 1 Here is the graph of y = sin x o. Click once to see the graph of y = sin 2x o. y = sin 2x o Notice the following points on the curve. It passes through the origin. It has a maximum of 1 (the same as a normal graph). It has a minimum of –1. It has a period of 360 o ÷ 2 = 180 o. x y

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Click to continue. y = sin x o O 180 o 360 o 1 Here is the graph of y = sin x o. Click once to see the graph of y = sin 3x o. y = sin 3x o Notice the following points on the curve. It passes through the origin. It has a maximum of 1 (the same as a normal graph). It has a minimum of –1. x y It has a period of 360 o ÷ 3 = 120 o.

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Click to continue. y = sin x o O 180 o 360 o 1 Here is the graph of y = sin x o. Click once to see the graph of y = sin ½x o. y = sin ½ x o Notice the following points on the curve. It passes through the origin. It has a maximum of 1 (the same as a normal graph). It has a minimum of –1. It has a period of 360 o ÷ ½ = 720 o. x y 540 o 720 o

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Click to continue. O Period is (360 o ÷ b) 1 Here is the graph of y = sin bx o. y = sin bx o x y It still passes through the origin. The altitude (or height) is unaffected. The period is 360 o b. The period is 360 o ÷ b.

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y = 4 sin x o O 180 o 1 x y This is the graph of which function? y = sin 2x o y = sin 4x o 90 o 45 o 135 o

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Which of these diagrams shows the graph of y = sin 6x o ? x y O 180 o 360 o 1 x y O 90 o 180 o 1 x y O 60 o 120 o 1 x y O 45 o 90 o -0.5 0.5 90 o 30 o -0.5 0.5 1

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Click to continue. For y = sin bx o only the period is affected. The graph will now have a period of 360 o b. This is also true for y = cos bx o and y = tan bx o. 90 o 180 o 270 o 360 o O x y 1 y = cos x o 90 o 180 o 270 o 360 o O x y 1 45 o y = tan x o Here are the graphs of y = cos x o and y = tan x o.

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Click to continue. Here are some examples of the graphs of y = cos bx o. 90 o 180 o 270 o O Click for y = cos 2x o period = 360 o ÷ 2 = 180 o Click for y = cos 2 / 3 x o period = 360 o ÷ 2 / 3 = 540 o Click for y = cos ½ x o period = 360 o ÷ ½ = 720 o y = cos x o 360 o y 450 o 540 o 630 o 1 y 720 o

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90 o 180 o -45 o -90 o O x y 1 45 o Click to continue. Here are some examples of the graphs of y = tan bx o. y = tan x o 3 2 4 -2 -3 -4 Notice this point Click to see y = tan 2x o period = 180 o ÷ 2 = 90 o and 45 o ÷ 2 = 22.5 o y = tan 2x o Click to see y = tan ½x o period = 180 o ÷ ½ = 360 o and 45 o ÷ ½ = 90 o Notice this point y = tan ½x o

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We will now look at graphs of the form y = sin x o + c. Click to continue. For example: y = sin x o + 2, y = sin x o + 3 or y = sin x o – 1.

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You are already familiar with the basic graph of y = sin x o. There are some important points to remember. 360 o 1 90 o 270 o O 180 o Click to continue. y = sin x o x y

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Click to continue. y = sin x o O 180 o 360 o 1 Here is the graph of y = sin x o. Click once to see the graph of y = sin x o + 1. y = sin x o + 1 Notice the following points on the curve. It passes through the origin + 1 = (0, 1). It has a maximum of 1 + 1 = 2. It has a minimum of –1 + 1 = 0. It has a period of 360 o. x y 2 -2 3 -3 The whole graph has been moved up one unit.

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Click to continue. Here is the graph of y = cos x o. 90 o 180 o 270 o O y = cos x o 360 o x y 1 Click once to see the graph of y = cos x o – 1. y = cos x o – 1 The whole graph has been moved down one unit.

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90 o 180 o 270 o 360 o O x y 1 45 o Click to continue. Here is the graph of y = tan x o. 450 o y = tan x o 3 2 4 -2 Notice this point Click once to see the graph of y = tan x o + 2. The whole graph has been moved up two units. y = tan x o + 2

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y = -3 sin x o O 720 o 1 x y This is the graph of which function? y = sin x o + 2 y = sin x o – 2 360 o 180 o 540 o 2 3 -2 -3

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Which of these diagrams shows the graph of y = cos x o + 2? x y O 180 o 360 o 2 -2 4 -4 x y O 360 o 2 -2 x y O 360 o 6 -2 2 4 540 o 180 o 540 o y O 180 o 360 o 2 3 1 x

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We will now look at graphs of the form y = a sin bx o + c, y = a cos bx o + c and y = a tan bx o + c. Click to continue. For example: y = 2 sin 3x o – 1, y = ½ cos 4x o + 3 or y = ¾ tan ¼x o – 12.

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Let us look at the graph of y = 2 sin 3x o – 1. Begin by considering the simple curve of y = sin x o. 180 o 540 o 360 o x y O Now, think on the graph of y = 2 sin x o : the 2 will double the height. The graph of y = 2 sin 3x o : the 3 makes the period as long (360 o ÷ 3 = 120 o ) 1 2 -2 -3 120 o Finally, y = 2 sin 3x o – 1, where the –1 moves the whole graph down one unit. y = 2 sin 3x o – 1 Click to continue.

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Look at this graph. What function does it show? 180 o y O 1 2 -2 -3 x 360 o 90 o 270 o 2. Next, look at the height. Maximum of 0.5 Minimum of –2.5 Therefore, the height is 3 units. Normally, a COSINE graph has a height of 2. Therefore the height has been multiplied by 3 ÷ 2 = 1.5 1. First, decide on the type. 3. Now, consider the period. The first complete wave finishes here. This means the period is 180 o so 360 o ÷ 180 o = 2. 4. Finally, find out how much it has been moved down (or up). This is the middle of the wave and it has been moved 1 unit down from the x-axis. It must be a COSINE graph because the first bump is on the y-axis. a = 1.5 b = 2 c = - 1 Therefore, we get –1. y = 1.5 cos 2x o - 1 Click to continue.

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Which of these graphs shows the function y = 2 sin 3x o + 1? x y O 180 o 360 o 1 x y O 360 o 2 -2 x y O 360 o 6 -2 2 4 1080 o 720 o 180 o 540 o y O 120 o 240 o 2 3 1 x 2 -3-2

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