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Trigonometric Graphs Written by B. Willox (Bridge of Don Academy) for Fourth Year – Credit Level 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 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 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 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 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 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 x y This is the graph of which function? y = 15 sin x o y = sin x o + 15

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WRONG! Try again.

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Well Done! Click to continue

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Which of these diagrams shows the graph of y = 7 sin x o ? x y O 180 o 360 o 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 o 180 o

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WRONG! The height (or altitude) is too big. Try again.

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Well Done! Click to continue

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WRONG! The period is too long. Try again.

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WRONG! The period is too short. Try again.

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WRONG! The period is too short. Try again.

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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 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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WRONG! Remember, a normal SINE graph has a period of 360 o. Try again.

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Well Done! Click to continue

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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 o 30 o

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WRONG! Remember, the period of a normal SINE graph is 360 o. Try again.

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Well Done! Click to continue

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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 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 = 2. It has a minimum of –1 + 1 = 0. It has a period of 360 o. x y 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 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 – o 180 o 540 o

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WRONG! Remember, a normal SINE graph has a height of 2 (from –1 to 1). Try again.

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Well Done! Click to continue

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

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WRONG! This is the graph of y = sin x o + 2. Try again.

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WRONG! This is the graph of y = cos x o + 1. Try again.

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Well Done! Click to continue

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WRONG! This is the graph of y = 2 cos x o + 3. Try again.

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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 ) 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 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 = 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 = 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 o 720 o 180 o 540 o y O 120 o 240 o x

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WRONG! This is the graph of y = 2 sin 2x o + 1. Try again.

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WRONG! This is the graph of y = cos x o + 1. Try again.

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WRONG! This is the graph of y = 4 cos ½x o + 1. Try again.

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Well Done! Click to continue

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Trigonometric Graphs Presentation is complete. Then go to and select The graphs of sin, cos and tan from the bottom of the screen. Then scroll down and click on the red boxes for “Recognize functions 3” and/or “Recognize graphs 3”.http://www.univie.ac.at/future.media/moe/galerie/trig/trig.htmlThe graphs of sin, cos and tan Want to try some harder questions?

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