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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Symmetry with respect to a point A graph is said to be symmetric with respect to a point Q if to each point P on the graph, we can find point P’ on the same graph, such that Q is the midpoint of the segment joining P and P’. Symmetry with respect to the axis or line A graph is said to be symmetric with respect to a line if the reflection (mirror image) about the line of every point on the graph is also on the graph The line is known as the line of symmetry.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Two points are symmetric with respect to the y – axis if and only if their x – coordinates are additive inverses and they have the same y – coordinate.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Two points are symmetric with respect to the x – axis if and only if their y –coordinates are additive inverses and they have the same x – coordinate.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Two points are symmetric with respect to the origin if and only if both their x – and y – coordinates are additive inverses of each other. Imagine sticking a pin in the given figure at the origin and then rotating the figure at 180 0. Points P and P 1 would be interchanged. The entire figure would look exactly as it did before rotating.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 A function is an even function when f(-x) = f(x) for all x in the domain of f. This is a function symmetric with respect to the y – axis. A function is an odd function when f(-x) = - f(x) for all x in the domain of f. This is a function symmetric with respect to the origin.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Graph of the Sine Function To sketch the graph of y = sin x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 0010sin x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = sin x
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Cosine Function Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 Graph of the Cosine Function To sketch the graph of y = cos x first locate the key points. These are the maximum points, the minimum points, and the intercepts. 1001cos x 0x Then, connect the points on the graph with a smooth curve that extends in both directions beyond the five points. A single cycle is called a period. y x y = cos x
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 6. The cycle repeats itself indefinitely in both directions of the x-axis. Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 1. The domain is the set of real numbers. 5. Each function cycles through all the values of the range over an x-interval of. 2. The range is the set of y values such that.
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Example: y = 3 cos x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 y x Example: Sketch the graph of y = 3 cos x on the interval [– , 4 ]. Partition the interval [0, 2 ] into four equal parts. Find the five key points; graph one cycle; then repeat the cycle over the interval. maxx-intminx-intmax 30-303 y = 3 cos x 22 0x (0, 3) (, 0) (, 3) (, –3)
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Amplitude Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function. amplitude = |a| If |a| > 1, the amplitude stretches the graph vertically. If 0 < |a| < 1, the amplitude shrinks the graph vertically. If a < 0, the graph is reflected in the x-axis. y x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x y = 2 sin x y = sin x
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Period of a Function Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 y x period: 2 period: The period of a function is the x interval needed for the function to complete one cycle. For b 0, the period of y = a sin bx is. For b 0, the period of y = a cos bx is also. If 0 < b < 1, the graph of the function is stretched horizontally. If b > 1, the graph of the function is shrunk horizontally. y x period: 2 period: 4
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Graph y = f(-x) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13 y x y = cos (–x) Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the graph of y = sin (–x). Use the identity sin (–x) = – sin x The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. Example 2: Sketch the graph of y = cos (–x). Use the identity cos (–x) = cos x The graph of y = cos (–x) is identical to the graph of y = cos x. y x y = sin x y = sin (–x) y = cos (–x)
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14 Steps in Graphing y = a sin bx and y = a cos bx. 4. Apply the pattern, then graph. 3. Find the intervals. 2. Find the period =. 1. Identify the amplitude =.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15 y = a cos bx y = a sin bx
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Example: y = 2 sin(-3x) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16 y x 0 20 –2 0y = –2 sin 3x 0 x Example: Sketch the graph of y = 2 sin (–3x). Rewrite the function in the form y = a sin bx with b > 0 amplitude: |a| = |–2| = 2 Calculate the five key points. (0, 0) (, 0) (, 2) (, -2) (, 0) Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x period: 2 2 3 =
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Tangent Function Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 y x Graph of the Tangent Function 2. range: (– , + ) 3. period: 4. vertical asymptotes: 1. domain : all real x Properties of y = tan x period: To graph y = tan x, use the identity. At values of x for which cos x = 0, the tangent function is undefined and its graph has vertical asymptotes.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18 Steps in Graphing y = a tan bx. 1. Determine the period. 2. Locate two adjacent vertical asymptotes by solving for x: 3. Sketch the two vertical asymptotes found in Step 2. 4. Divide the interval into four equal parts. 5. Evaluate the function for the first – quarter point, midpoint, and third - quarter point, using the x – values in Step 4. 6. Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary.
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 2. Find consecutive vertical asymptotes by solving for x: 4. Sketch one branch and repeat. Example: Find the period and asymptotes and sketch the graph of Vertical asymptotes: 3. Plot several points in 1. Period of y = tan x is . y x
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20 2. Find consecutive vertical asymptotes by solving for x: 4. Sketch one branch and repeat. Example: Find the period and asymptotes and sketch the graph of Vertical asymptotes: 3. Divide - to into four equal parts. 1. Period of y = tan x is. of Period is y x
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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21 Graph 1. Period is or 4 . 2. Vertical asymptotes are 3. Divide the interval - 2 to 2 into four equal parts. y x x = - 2 x = 2
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