Graphs of Trigonometric Functions

Symmetry with respect to a point 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. 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’. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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 1800. Points P and P1 would be interchanged. The entire figure would look exactly as it did before rotating. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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. -1 1 sin x x 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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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. 1 -1 cos x x 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 = cos x y x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Cosine Function

Properties of Sine and Cosine Functions The graphs of y = sin x and y = cos x have similar properties: 1. The domain is the set of real numbers. 2. The range is the set of y values such that . 3. The maximum value is 1 and the minimum value is –1. 4. The graph is a smooth curve. 5. Each function cycles through all the values of the range over an x-interval of . 6. The cycle repeats itself indefinitely in both directions of the x-axis. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

If |a| > 1, the amplitude stretches the graph vertically. 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 = 2 sin x y = sin x y = sin x y = – 4 sin x reflection of y = 4 sin x y = 4 sin x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Amplitude

If b > 1, the graph of the function is shrunk horizontally. 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 b > 1, the graph of the function is shrunk horizontally. y x period: 2 period: If 0 < b < 1, the graph of the function is stretched horizontally. y x period: 4 period: 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Period of a Function

Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the graph of y = sin (–x). The graph of y = sin (–x) is the graph of y = sin x reflected in the x-axis. y x y = sin (–x) Use the identity sin (–x) = – sin x y = sin x Example 2: Sketch the graph of y = cos (–x). The graph of y = cos (–x) is identical to the graph of y = cos x. y x Use the identity cos (–x) = cos x y = cos (–x) y = cos (–x) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graph y = f(-x)

Identify the amplitude = . Steps in Graphing y = a sin bx and y = a cos bx. Identify the amplitude = . 2. Find the period = . 3. Find the intervals. 4. Apply the pattern, then graph. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

y = a cos bx y = a sin bx Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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. max x-int min 3 -3 y = 3 cos x 2  x y x (0, 3) ( , 3) ( , 0) ( , 0) ( , –3) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: y = 3 cos x

Use the identity sin (– x) = – sin x: Example: Sketch the graph of y = 2 sin (–3x). Rewrite the function in the form y = a sin bx with b > 0 Use the identity sin (– x) = – sin x: y = 2 sin (–3x) = –2 sin 3x period: 2 3 = amplitude: |a| = |–2| = 2 Calculate the five key points. 2 –2 y = –2 sin 3x x y x ( , 2) (0, 0) ( , 0) ( , 0) ( , -2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: y = 2 sin(-3x)

Graph of the Tangent Function 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. y x Properties of y = tan x 1. domain : all real x 2. range: (–, +) 3. period:  4. vertical asymptotes: period: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Tangent Function

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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Example: Find the period and asymptotes and sketch the graph of 1. Period of y = tan x is  . 2. Find consecutive vertical asymptotes by solving for x: Vertical asymptotes: 3. Plot several points in 4. Sketch one branch and repeat. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

Example: Find the period and asymptotes and sketch the graph of 1. Period of y = tan x is . of Period ® is 2. Find consecutive vertical asymptotes by solving for x: Vertical asymptotes: 3. Divide - to  into four equal parts. 4. Sketch one branch and repeat. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

2. Vertical asymptotes are x = - 2 x = 2 y x Graph 1. Period is or 4. 2. Vertical asymptotes are 3. Divide the interval - 2 to 2 into four equal parts. Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

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