MAC 1114 Module 4 Graphs of the Circular Functions
Learning Objectives Upon completing this module, you should be able to: 1. Recognize periodic functions. 2. Determine the amplitude and period, when given the equation of a periodic function. 3. Find the phase shift and vertical shift, when given the equation of a periodic function. 4. Graph sine and cosine functions. 5. Graph cosecant and secant functions. 6. Graph tangent and cotangent functions. 7. Interpret a trigonometric model. 2
Graphs of the Circular Functions There are three major topics in this module: - Graphs of the Sine and Cosine Functions - Translations of the Graphs of the Sine and Cosine Functions - Graphs of the Other Circular Functions 3
Introduction to Periodic Function A periodic function is a function f such that f(x) = f(x + np), for every real number x in the domain of f, every integer n, and some positive real number p. The smallest possible positive value of p is the period of the function. 4
Example of a Periodic Function 5
Example of Another Periodic Function 6
What is the Amplitude of a Periodic Function? The amplitude of a periodic function is half the difference between the maximum and minimum values. The graph of y = a sin x or y = a cos x, with a, will have the same shape as the graph of y = sin x or y = cos x, respectively, except the range will be [ a, a ]. The amplitude is a. 7
How to Graph y = 3 sin(x)? x π/2 π 3π/2 π sin x 1 1 3sin x 3 3 Note the difference between sin x and 3sin x. What is the difference? 8
How to Graph y = sin(2x)? The period is 2π/2 = π. The graph will complete one period over the interval [, π]. The endpoints are and π, the three middle points are: Plot points and join in a smooth curve. 9
How to Graph y = sin(2x)? (Cont.) Note the difference between sin x and sin 2x. What is the difference? 1
Period of a Periodic Function Based on the previous example, we can generalize the following: For b >, the graph of y = sin bx will resemble that of y = sin x, but with period 2π/b. The graph of y = cos bx will resemble that of y = cos x, with period 2π/b. 11
How to Graph y = cos (2x/3) over one period? The period is 3π. Divide the interval into four equal parts. Obtain key points for one period. x 3π/4 3π/2 9π/4 3π 2x/3 π/2 π 3π/2 2π cos 2x/3 1 1 1 12
How to Graph y = cos(2x/3) over one period? (Cont.) The amplitude is 1. Join the points and connect with a smooth curve. 13
Guidelines for Sketching Graphs of Sine and Cosine Functions To graph y = a sin bx or y = a cos bx, with b >, follow these steps. Step 1 Find the period, 2π/b. Start with on the x-axis, and lay off a distance of 2π/b. Step 2 Divide the interval into four equal parts. Step 3 Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and x-intercepts. 14
Guidelines for Sketching Graphs of Sine and Cosine Functions Continued Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude a. Step 5 Draw the graph over additional periods, to the right and to the left, as needed. 15
How to Graph y = 2 sin(4x)? Step 1 Period = 2π/4 = π/2. The function will be graphed over the interval [, π/2]. Step 2 Divide the interval into four equal parts. Step 3 Make a table of values x π/8 π/4 3π/8 π/2 4x π/2 π 3π/2 2π sin 4x 1 1 2 sin 4x 2 2 16
How to Graph y = 2 sin(4x)? (Cont.) Plot the points and join them with a sinusoidal curve with amplitude 2. 17
What is a Phase Shift? In trigonometric functions, a horizontal translation is called a phase shift. In the equation the graph is shifted π/2 units to the right. 18
How to Graph y = sin (x π/3) by Using Horizontal Translation or Phase Shift? Find the interval for one period. Divide the interval into four equal parts. 19
How to Graph y = sin (x π/3) by Using Horizontal Translation or Phase Shift? (Cont.) x x π/3 sin (x π/3) π/3 5π/6 π/2 1 4π/3 π 11π/6 3π/2 1 7π/3 2π 2
How to Graph y = 3 cos(x + π/4) by Using Horizontal Translation or Phase Shift? Find the interval. Divide into four equal parts. 21
How to Graph y = 3 cos(x + π/4) by Using Horizontal Translation or Phase Shift? x x + π/4 cos(x + π/4) 3 cos (x + π/4) π/4 1 3 π/4 π/2 3π/4 π 1 3 5π/4 3π/2 7π/4 2π 1 3 22
How to Graph y = 2 2 sin 3x by Using Vertical Translation or Vertical Shift? The graph is translated 2 units up from the graph y = 2 sin 3x. x π/6 π/3 π/2 2π/3 3x π/2 π 3π/2 2π 2 sin 3x 2 2 2 2 sin 3x 2 2 4 2 23
How to Graph y = 2 2 sin 3x by Using Vertical Translation or Vertical Shift? (Cont.) Plot the points and connect. 24
Further Guidelines for Sketching Graphs of Sine and Cosine Functions Method 1: Follow these steps. Step 1 Step 2 Step 3 Find an interval whose length is one period 2π/b by solving the three part inequality b(x d) 2π. Divide the interval into four equal parts. Evaluate the function for each of the five x-values resulting from Step 2. The points will be maximum points, minimum points, and points that intersect the line y = c (middle points of the wave.) 25
Further Guidelines for Sketching Graphs of Sine and Cosine Functions (Cont.) Step 4 Plot the points found in Step 3, and join them with a sinusoidal curve having amplitude a. Step 5 Draw the graph over additional periods, to the right and to the left, as needed. Method 2: First graph the basic circular function. The amplitude of the function is a, and the period is 2π/b. Then use translations to graph the desired function. The vertical translation is c units up if c > and c units down if c <. The horizontal translation (phase shift) is d units to the right if d > and d units to the left if d <. 26
How to Graph y = 1 + 2 sin (4x + π)? Write the expression in the form c + a sin b(x d) by rewriting 4x + π as Step 2: Divide the interval. Step 1 Step 3 Table 27
How to Graph y = 1 + 2 sin (4x + π)?(cont.) x π/4 π/8 π/8 π/4 x + π/4 π/8 π/4 3π/8 π/2 4(x + π/4) π/2 π 3π/2 2π sin 4(x + π/4) 1 1 2 sin 4(x + π/4) 2 2 2 1 + 2sin(4x + π) 1 1 1 3 1 28
How to Graph y = 1 + 2 sin (4x + π)? (Cont.) Steps 4 and 5 Plot the points found in the table and join then with a sinusoidal curve. 29
Let s Take a Look at Other Circular Functions. 3
Cosecant Function 31
Secant Function 32
Guidelines for Sketching Graphs of Cosecant and Secant Functions To graph y = csc bx or y = sec bx, with b >, follow these steps. Step 1 Graph the corresponding reciprocal function as a guide, using a dashed curve. To Graph y = a csc bx y = a sec bx Use as a Guide y = a sin bx y = cos bx 33
Guidelines for Sketching Graphs of Cosecant and Secant Functions Continued Step 2 Sketch the vertical asymptotes. - They will have equations of the form x = k, where k is an x-intercept of the graph of the guide function. Step 3 Sketch the graph of the desired function by drawing the typical U-shapes branches between the adjacent asymptotes. - The branches will be above the graph of the guide function when the guide function values are positive and below the graph of the guide function when the guide function values are negative. 34
How to Graph y = 2 sec(x/2)? Step 1: Graph the corresponding reciprocal function y = 2 cos (x/2). The function has amplitude 2 and one period of the graph lies along the interval that satisfies the inequality Divide the interval into four equal parts. 35
How to Graph y = 2 sec(x/2)? (Cont.) Step 2: Sketch the vertical asymptotes. These occur at x- values for which the guide function equals, such as x = 3π, x = 3π, x = π, x = 3π. Step 3: Sketch the graph of y = 2 sec x/2 by drawing the typical U-shaped branches, approaching the asymptotes. 36
Tangent Function 37
Cotangent Function 38
Guidelines for Sketching Graphs of Tangent and Cotangent Functions To graph y = tan bx or y = cot bx, with b >, follow these steps. Step 1 Determine the period, π/b. To locate two adjacent vertical asymptotes solve the following equations for x: 39
Guidelines for Sketching Graphs of Tangent and Cotangent Functions continued Step 2 Sketch the two vertical asymptotes found in Step 1. Step 3 Divide the interval formed by the vertical asymptotes into four equal parts. Step 4 Evaluate the function for the first-quarter point, midpoint, and third-quarter point, using the x-values found in Step 3. Step 5 Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary. 4
How to Graph y = tan(2x)? Step 1: The period of the function is π/2. The two asymptotes have equations x = π/4 and x = π/4. Step 2: Sketch the two vertical asymptotes found. x = ± π/4. Step 3: Divide the interval into four equal parts. This gives the following key x-values. First quarter: π/8 Middle value: Third quarter: π/8 41
How to Graph y = tan(2x)? (Cont.) Step 4: Evaluate the function x π/8 π/8 2x π/4 π/4 tan 2x 1 1 Step 5: Join the points with a smooth curve, approaching the vertical asymptotes. Indicate additional asymptotes and periods of the graph as necessary. 42
How to Graph y = tan(2x)? (Cont.) Every y value for this function will be 2 units more than the corresponding y in y = tan x, causing the graph to be translated 2 units up compared to y = tan x. 43
We have learned to: What have we learned? 1. Recognize periodic functions. 2. Determine the amplitude and period, when given the equation of a periodic function. 3. Find the phase shift and vertical shift, when given the equation of a periodic function. 4. Graph sine and cosine functions. 5. Graph cosecant and secant functions. 6. Graph tangent and cotangent functions. 7. Interpret a trigonometric model. 44
Credit Some of these slides have been adapted/modified in part/whole from the slides of the following textbook: Margaret L. Lial, John Hornsby, David I. Schneider, Trigonometry, 8th Edition 45