June 6 Math 1113 sec 002 Summer 2014
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1 June 6 Math 1113 sec 002 Summer 2014 Sec. 6.4 Plotting f (x) = a sin(bx c) + d or f (x) = a cos(bx c) + d Amplitude is a. If a < 0 there is a reflection in the x-axis. The fundamental period is The phase shift is c c < 0). b T = 2π b. units (to the right if c > 0 and the left if There is a vertical shift d units (up if d > 0 and down if d < 0). () June 4, / 81
2 Basic Plots Figure: One period of basic sine and cosine plots. () June 4, / 81
3 Pulling it all Together! Plot two full periods of the function f (x) = a sin(bx c) + d (or f (x) = a cos(bx c) + d). Carry out each of the following steps: Identify the amplitude and determine if there is an x-axis reflection. Identify the period. Find the length of one fourth of the period. Identify any phase shift with its direction. Identify end points and points that divide the period into four equal parts. Identify any vertical shift with its direction. Use the basic plot of y = sin x or y = cos x to get the profile. () June 4, / 81
4 f (x) = 2 4 cos ( πx π 2 ) Identify the amplitude and period. Determine if there is a reflection in the x-axis. Find the length of one quarter period. () June 4, / 81
5 f (x) = 2 4 cos ( πx π 2 ) Identify any phase shift and vertical shift along with direction. Determine the end points and points that divide the period into four parts. () June 4, / 81
6 f (x) = 2 4 cos ( πx π 2 Plot two periods of its graph. ) () June 4, / 81
7 Basic Plot f (x) = tan x Recall that the domain of the tangent function is {x x kπ 2, k = ±1, ±3, ±5,...} Remark 1: The graph of the tangent function has vertical asymptotes corresponding to each value that is not in the domain. That is, for every odd integer k, x = kπ 2 is a vertical asymptote to the graph of f (x) = tan x. Remark 2: The tangent function has period π, so the basic profile between a pair of adjacent asymptotes is repeated. () June 4, / 81
8 Basic Plot f (x) = tan x We can plot one period between asymptotes x = π 2 and x = π 2 by considering a few values: x π π π π π π tan x () June 4, / 81
9 Basic Plot f (x) = tan x Figure: Plot of several periods of f (x) = tan x. () June 4, / 81
10 Basic Plot f (x) = cot x Recall that the domain of the cotangent function is {x x kπ, k = 0, ±1, ±2, ±3,...} Remark 1: The graph of the cotangent function has vertical asymptotes corresponding to each value that is not in the domain. That is, for every integer k, x = kπ is a vertical asymptote to the graph of f (x) = cot x. Remark 2: The cotangent function has period π, so the basic profile between a pair of adjacent asymptotes is repeated. () June 4, / 81
11 Basic Plot f (x) = cot x To plot f (x) = cot(x) we can use the identities (cofunction & odd symmetry) ( π ) ( ( cot(x) = tan 2 x = tan x π )) ( = tan x π ). 2 2 The plot of the cotangent is obtained from that of the tangent by shifting π 2 units to the right and reflecting in the x-axis. () June 4, / 81
12 Basic Plot f (x) = cot x Figure: Plot of several periods of f (x) = cot x. () June 4, / 81
13 f (x) = a tan(bx c) + d or f (x) = a cot(bx c) + d If the number a = 1, we say that f has no vertical stretch. Definition: If a ±1 is any other nonzero number, then the graph of f above has a vertical stretch by a factor of a units. If a < 0, the graph has a reflection in the x-axis. () June 4, / 81
14 Example Determine if the function has any vertical stretch. If so, determine how many units it is. Does the graph have a reflection in the x-axis? ( (a) f (x) = 4 tan 4x π ) +1 2 (b) f (x) = 2 5 cot ( πx ) 2 (c) ( f (x) = cot 2x + π ) +3 3 () June 4, / 81
15 f (x) = a tan(bx c) + d or f (x) = a cot(bx c) + d Theorem: For any positive number b, the fundamental period of f above is T = π b. If f is the tangent function, then its asymptotes will be x = kπ 2b + c, for k = ±1, ±3, ±5,... b If f is the cotangent function, then its asymptotes will be x = kπ b + c, for k = 0, ±1, ±2, ±3,... b () June 4, / 81
16 Example Determine the period of each function and find four adjacent asymptotes. 1 ( (a) f (x) = 4 tan 4x π ) asymp. for tan(bx c): x = kπ 2b + c, for k = ±1, ±3, ±5,... b () June 4, / 81
17 Example 2 (b) ( f (x) = cot 2x + π ) asymp. for cot(bx c): x = kπ b + c, for k = 0, ±1, ±2, ±3,... b () June 4, / 81
18 f (x) = a tan(bx c) + d or f (x) = a cot(bx c) + d If b = 1 we say that f does not have a horizontal stretch. Definition: For any positive number b 1 the graph of f above has a horizontal stretch by a factor of b units. () June 4, / 81
19 Example Determine if the function has any horizontal stretch. If so, determine its factor. (a) ( f (x) = 4 tan 4x π ) +1 2 (b) f (x) = 2 5 cot ( πx ) 2 (c) ( f (x) = cot 2x + π ) +3 3 () June 4, / 81
20 f (x) = a tan(bx c) + d or f (x) = a cot(bx c) + d Definition: If c is a nonzero number, then the graph of f has a horizontal shift (also called a phase shift) of c b units to the right if c > 0 and to the left if c < 0. () June 4, / 81
21 Example Determine the horizontal shift if any, and indicate its direction. (a) ( f (x) = 4 tan 4x π ) +1 2 (b) f (x) = 2 5 cot ( πx ) 2 (c) ( f (x) = cot 2x + π ) +3 3 () June 4, / 81
22 f (x) = a tan(bx c) + d or f (x) = a cot(bx c) + d Definition: If d is a nonzero number, then the function f has a vertical shift of d units up if d > 0 and down if d < 0. () June 4, / 81
23 Example Determine the vertical shift if any, and indicate its direction. (a) ( f (x) = 4 tan 4x π ) +1 2 (b) f (x) = 2 5 cot ( πx ) 2 (c) ( f (x) = cot 2x + π ) +3 3 () June 4, / 81
24 Pulling it all Together! Plot two full periods of the function f (x) = a tan(bx c) + d (or f (x) = a cot(bx c) + d). Carry out each of the following steps: Identify any vertical stretch and determine if there is an x-axis reflection. Identify the period and any horizontal stretch. Identify any horizontal (a.k.a phase) shift with its direction. Find at least three adjacent asymptotes Tangent: x = kπ 2b + c, for k = ±1, ±3, ±5,... b Cotangent: x = kπ b + c, for k = 0, ±1, ±2, ±3,... b Identify any vertical shift with its direction. Use the basic plot of y = tan x or y = cot x to get the profile. () June 4, / 81
25 f (x) = 2 tan ( x + π 4) 1 Identify any vertical stretch and any x-axis reflection. Identify the period and any horizontal stretch. () June 4, / 81
26 f (x) = 2 tan ( x + π 4) 1 Identify any horizontal shift along with its direction. And find the equations of at least three adjacent vertical asymptotes. Determine any vertical shift along with its direction. () June 4, / 81
27 f (x) = 2 tan ( x + π 4) 1 Plot two full periods. () June 4, / 81
28 f (x) = 2 3 cot(πx π) Identify any vertical stretch and any x-axis reflection. Identify the period and any horizontal stretch. () June 4, / 81
29 f (x) = 2 3 cot(πx π) Identify any horizontal shift along with its direction. And find the equations of at least three adjacent vertical asymptotes. Determine any vertical shift along with its direction. () June 4, / 81
30 f (x) = 2 3 cot(πx π) Plot two full periods. () June 4, / 81
31 PLotting the Secant or Cosecant Use the relationship csc x = 1 sin x to plot f (x) = csc x. () June 4, / 81
32 PLotting the Secant or Cosecant Use the relationship sec x = 1 cos x to plot f (x) = sec x. () June 4, / 81
33 PLotting the Secant or Cosecant To plot f (x) = a csc(bx c) + d Plot the related function g(x) = a sin(bx c) + d. At each point where g passes through the horizontal line y = d, sketch in a vertical asymptote. The graphs of f and g will touch at the maximum and minimum points. Plot the profile of f (x) between each of the asymptotes. () June 4, / 81
34 PLotting the Secant or Cosecant To plot f (x) = a sec(bx c) + d Plot the related function g(x) = a cos(bx c) + d. At each point where g passes through the horizontal line y = d, sketch in a vertical asymptote. The graphs of f and g will touch at the maximum and minimum points. Plot the profile of f (x) between each of the asymptotes. () June 4, / 81
35 Some Terminology for PLotting the Secant or Cosecant To plot f (x) = a csc(bx c) + d or f (x) = a sec(bx c) + d Vertical Stretch by a factor of a if a ±1 Reflection in x-axis if a < 0 Horizontal Stretch by a factor of b if b 1 Vertical Shift d units (up if d > 0 and down if d < 0) Horizontal Shift c b units (right if c > 0 and left if c < 0) () June 4, / 81
36 f (x) = 2 csc(2x π) + 1 Identify the function g(x). Find its amplitude. Find its period, and length of a quarter period. () June 4, / 81
37 f (x) = 2 csc(2x π) + 1 Identify any phase shift and vertical shift along with direction for the function g(x). Find end points and quarter points for the periods. () June 4, / 81
38 f (x) = 2 csc(2x π) + 1 Plot two periods of its graph. () June 4, / 81
39 f (x) = 2 sec ( ) πx 2 Identify the function g(x). Find its amplitude. Find its period, and length of a quarter period. () June 4, / 81
40 f (x) = 2 sec ( ) πx 2 Identify any phase shift and vertical shift along with direction for the function g(x). Find end points and quarter points for the periods. () June 4, / 81
41 f (x) = 2 sec ( ) πx 2 Plot two periods of its graph. () June 4, / 81
42 Section 6.5: Inverse Trigonometric Functions Question: If someone asks what is the sine of π 6? we can respond with the answer (from memory or perhaps using a calculator) 1 2. What if the question is reversed? What if someone asks What angle has a sine value of 1 2? () June 4, / 81
43 Restricting the Domain of sin(x) Figure: To define an inverse sine function, we start by restricting the domain of sin(x) to the interval [ π 2, ] π 2 () June 4, / 81
44 The Inverse Sine Function (a.k.a. arcsine function) Definition: For x in the interval [ 1, 1] the inverse sine of x is denoted by either and is defined by the relationship sin 1 (x) or arcsin(x) y = sin 1 (x) x = sin(y) where π 2 y π 2. The Domain of the Inverse Sine is 1 x 1. The Range of the Inverse Sine is π 2 y π 2. () June 4, / 81
45 Notation Warning! Caution: We must remember not to confuse the superscript 1 notation with reciprocal. That is sin 1 (x) 1 sin(x). If we want to indicate a reciprocal, we should use parentheses or trigonometric identities 1 sin(x) = (sin(x)) 1 or write 1 sin(x) = csc(x). () June 4, / 81
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