Graphs of Trigonometric Functions Symmetry with respect to

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Graphs of Trigonometric Functions

Graphs of Trigonometric Functions

Symmetry with respect to the axis or line A graph is said to be

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. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 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’. 2

Two points are symmetric with respect to the y – axis if and only

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. 3

Two points are symmetric with respect to the x – axis if and only

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. 4

Two points are symmetric with respect to the origin if and only if both

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 P 1 would be interchanged. The entire figure would look exactly as it did before rotating. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5

A function is an even function when f(-x) = f(x) for all x in

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. 6

Graph of the Sine Function To sketch the graph of y = sin x

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. x 0 sin x 0 1 0 -1 0 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 = sin x y x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

Graph of the Cosine Function To sketch the graph of y = cos x

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. x 0 cos x 1 0 -1 0 1 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 y = cos x x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

Properties of Sine and Cosine Functions The graphs of y = sin x and

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. 9

Example: Sketch the graph of y = 3 cos x on the interval [–

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. x y = 3 cos x (0, 3) y 0 3 max 0 -3 0 2 3 x-int min x-int max ( , 3) x ( ( , 0) ( , – 3) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

The amplitude of y = a sin x (or y = a cos x)

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 y = 2 sin x x y= sin x y = – 4 sin x reflection of y = 4 sin x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. y = sin x y = 4 sin x 11

The period of a function is the x interval needed for the function to

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 they function is shrunk horizontally. period: 2 x If 0 < b < 1, the graph of the function is stretched horizontally. y period: 2 x period: 4 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

Use basic trigonometric identities to graph y = f (–x) Example 1: Sketch the

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 = sin (–x) y Use the identity sin (–x) = – sin x 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 Use the identity x cos (–x) = cos x y = cos (–x) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

Steps in Graphing y = a sin bx and y = a cos bx.

Steps in Graphing y = a sin bx and y = a cos bx. 1. 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. 14

y = a cos bx y = a sin bx Copyright © by Houghton

y = a cos bx y = a sin bx Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 15

Example: Sketch the graph of y = 2 sin (– 3 x). Rewrite the

Example: Sketch the graph of y = 2 sin (– 3 x). Rewrite the function in the form y = a sin bx with b > 0 y = 2 sin (– 3 x) = – 2 sin 3 x Use the identity sin (– x) = – sin x: period: 2 = 2 amplitude: |a| = |– 2| = 2 3 Calculate the five key points. x 0 y = – 2 sin 3 x 0 y – 2 0 ( , 2) x (0, 0) ( , -2) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

Graph of the Tangent Function To graph y = tan x, use the identity

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 Properties of y = tan x 1. domain : all real x 2. range: (– , + ) 3. period: x 4. vertical asymptotes: period: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17

Steps in Graphing y = a tan bx. 1. Determine the period . 2.

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. 18

Example: Find the period and asymptotes and sketch the graph y of 1. Period

Example: Find the period and asymptotes and sketch the graph y of 1. Period of y = tan x is . 2. Find consecutive vertical asymptotes by solving for x: x Vertical asymptotes: 3. Plot several points in 4. Sketch one branch and repeat. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19

Example: Find the period and asymptotes and sketch the graph y of 1. Period

Example: Find the period and asymptotes and sketch the graph y of 1. Period of y = tan x is ® Period of . is 2. Find consecutive vertical asymptotes by solving for x: 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. 20

x = - 2 y Graph 1. Period is x = 2 or 4.

x = - 2 y Graph 1. Period is x = 2 or 4. x 2. Vertical asymptotes are 3. Divide the interval - 2 to 2 into four equal parts. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 21

Reporters: Naco, Sheree Pascua, Nathaniel Patacsil, Demi Peña, Kevin Peñaflor, Jezer http: //webcache. googleusercontent.

Reporters: Naco, Sheree Pascua, Nathaniel Patacsil, Demi Peña, Kevin Peñaflor, Jezer http: //webcache. googleusercontent. com/search? q=cache: GWXyb. X 60 i 0 g. J: team. zobel. dlsu. edu. ph/sites/students/H 2/Lists/Announcements/Attac hments/94/Graphs%2520 of%2520 Sine, %2520 Cosine%2520 and%2520 Tangent%2520 Functions. ppt+graphs+of+trigonometric+functions+ppt& cd=3&hl=tl&ct=clnk&gl=ph