left endpoint: solve bx-c=0 ·Interval right endpoint: solve bx-c=2π ·Vertical **shifts** are caused by d Precalculus 4.5 **Graphs** of Sine and Cosine 7 Steps to **Graph** ·Find **amplitude** ·Find left endpoint ·bx-c=0 ·Find right endpoint ·bx-c=2π ·Find the period · 2π/b ·Find **key** points by: ·Dividing period-interval into four equal parts ·Add “period/4” starting/

! y=A ???[B(θ-h)]+k 2: **Graphing** Trigonometric Functions 2: Example for **Graphing** **Graph** y=csc( - /2)+1. The vertical **shift** is 1. Use this information to **graph** the function. **Amplitude** is 1. The period is 2 /1 or 2 . The phase **shift** -(- /2/1) or /2. 2: Example for **Graphing** YOU TRY! **Graph** y=csc(2 - /2)+1. 2: Example for **Graphing** YOU TRY! **Graph** y=csc(2 - /2)+1/

a.k.a discrete-time). **Amplitude** Modulation (AM) Frequency Modulation (FM) Phase Modulation (PM) **Amplitude** **Shift** **Keying** (ASK) Frequency **Shift** **Keying** (FSK) Phase **Shift** **Keying** (PSK) Digital Modulation Schemes Figure 4-8 WCB/McGraw-Hill The McGraw-Hill Companies, Inc., 1998 **Amplitude** Change Figure 4-9 WCB//i will start to vary/deviate from f c with the amount of. The conversion can be seen from the **graph** fcfc fifi vsvs 0 k f = Conversion gain Frequency Modulation (FM) Fact in FM : Instantaneous frequency f/

-2-- 30-60-90 45-45-90 **Graphs** of Functions Sine **Graphs** of Functions Cosine **Graphs** of Functions Tangent **Graphs** of Functions Sine Cosine **Graphs** of Functions Tan Cotangent **Graphs** of Functions Secant Cosecant Chapter 4 - Lesson 2 Transforming Trig Functions Essential Question: How can we use the **amplitude**, period, phase **shift** and vertical **shift** to transform the sine and cosine curves? **Key** Question: How do the values of A/

. The range of the cosine curve is ________________. OBJECTIVE: SWBAT use trigonometric **graphs** to define and interpret features such as domain, range, intercepts, periods, **amplitude**, phase **shifts**, vertical **shifts** and asymptotes. SWBAT to **graph** Sine and Cosine functions with and without **graphing** technology. EQ: What are the **key** points for the basic Sine and Cosine parent **graphs** & where do they come from? How do the values of “a/

new reference period into 4 equal parts to create new x values for the **key** points Adjust the y values of the **key** points by applying the **amplitude** (a) and the vertical **shift** (d) **Graph** **key** points and connect the dots **Key** Steps in **Graphing** Secant and Cosecant Identify the **key** points of your reciprocal **graph** (sine/cosine), note the original zeros, maximums and minimums Find the new period (2/

is it unitary? Quantum Fourier Transform And about that inverse QFT: It performs the inverse Fourier transform on the **amplitudes**! In Class Problem #1 Period Finding quantum oracle Problem: find in as few queries as possible Period Finding / [Long List] Pell’s equation [Hallgren 02] Hidden **shift** problems [van Dam, Hallgren, Ip 03] **Graph** traversal [CCDFGS 03] Spatial search [AA 03, CG 03/04, AKR 04] Element distinctness [Ambainis 03] Various **graph** problems [DHHM 04, MSS 03,…] Testing matrix multiplication [/

**Amplitude** and Period of Sine FUNctions **amplitude** (new word, old concept) – the maximum displacement from equilibrium. For y = asinx and y = acosx, the **amplitude** is a. Let’s **graph** y = 2sinx. First, we will **graph** y = sinx. Next, we will label the five new **key** points, and **graph** the/of your endpoints, then average that value with each endpoint.) **Key** Points (-π/4,0) (0,2) (π/4,0) (π/2,-2) (3π/4,0) The **graph**! Good news! Vertical **shifts** are easy! We just **shift** up or down after we are finished with everything else./

x is ). Five **Key** Points: Intercepts, Max, and Min. **Amplitude** = half the distance bet. the max. and min. (**amplitude** for y = sin x is 1.) **Graph** of y = cos x Cosine **graph** is periodic (period for y = cos x is ). Five **Key** Points: Intercepts, Max, and Min. **Amplitude** = half the distance bet. the max. and min. (**amplitude** for y = cos x is 1.) **Graphing**: **Amplitude** = Period = Phase **Shift** : solve equations and/

for y = cos x is ). Five **Key** Points: Intercepts, Max, and Min. **Amplitude** = half the distance bet. the max. and min. (**amplitude** for y = cos x is 1.) **Graphing**: **Amplitude** = Period = Divide the period by 4 to get four equal subintervals. Phase **Shift** : solve equations and. This will give you the left and right endpoints of a period. Vertical **Shift** = The **graph** of is a reflection in the/

, to find the period when B is not one, the period of the function is the following: 2π Steps for **Graphing** 1. “**Shift**” the **graph** up or down and draw in the new midline. 2. Mark off the **amplitude**. 3. Determine the period and scale your **graph** accordingly. 4. Mark **key** points and sketch the **graph**. *Be careful if the A is negative, make sure to flip/

stability. Also, PM is adaptable to data applications Examples of Phase **Shift** Bit Rate = Baud rate x Bits per Symbol PSK and QAM Phase **Shift** **Keying** (PSK) Most popular implementation of PM for data In BPSK (Binary PSK): one bit per phase change In QPSK: two bits per phase change (symbol) Quadrature **Amplitude** Modulation (QAM) Uses two AM carriers with 90o phase angle between/

The cycle will begin at 0 and end at π/2 There are five **key** points of a sinusoidal **graph**: The beginning of the cycle, the points where sin(x)= 0, and the local maximum and minimum. Since this **graph** is not **shifted** horizontally, the beginning points is (0,0). Notice that there are / + v ω = -π/2 (However we need ω > 0) Let’s remember that sin x is an odd function, where f(-x) = -f(x) So The **amplitude** is |-2| = 2 so the largest value of y is 2. The period is T = 2π/ω = φ = 0, v = 0, so there is no horizontal /

**amplitude** (plus the phase **shift**) and to select a Ymax at least one value higher than that value. Select an appropriate y-scl based on the size of your **amplitude**. A “1” value will probably be sufficient. Leave Xres = 1. Viewing the **Graph** Press the y= **key**. Check to be sure the “Plot 1” **key** is not highlighted. Enter the equation under Y1=. Press the **graph** **key**. SAMPLE **GRAPH**/

the period and **amplitude** Find the period and **amplitude** Describe the relationship between the **graphs** of f and g Describe the relationship between the **graphs** of f and g. Consider **amplitudes**, periods, and **shifts**. p. 294 #15 Describe the relationship between the **graphs** of f and g Describe the relationship between the **graphs** of f and g. Consider **amplitudes**, periods, and **shifts**. p. 294 #21 Sketch the **graphs** of f and/

interval on the x-axis. 5.Divide the interval into fourths to plot “**key** points”. 6.**Graph** one period. Extend if necessary. 4.5 – **Graphs** of Sine and Cosine **Graph** the equation y = 3 sin(2x − π) **amplitude**: period: phase **shift**: interval: 3 = π π2π2π 4.5 – **Graphs** of Sine and Cosine **Graph** the equation y = **amplitude**: period: p. s.: interval: 4π4π2π2π = 2π(2)= 4π = −π(2)= −2/

is needed 136 Frequency **Shift** **Keying** (FSK) Carrier Modulating signal FSK signal 137 FSK (cont’d) The frequency of the FSK signal changes abruptly from one that is higher than that of the carrier to one that is lower. Note that the **amplitude** of the FSK signal remains/, 8.65, etc. These points are useful for determining the deviation and the value of k f of an FM modulator system. 146 **Graph** of Bessel Functions 147 FM Side-Bands Each (J) value in the table gives rise to a pair of side- frequencies. The higher /

1 2 -2 y = A sin (Bx – C)**amplitude** = | A | period = 2 B phase **shift** = C (right/left) B y = sin (x) y = 2 sin (x) y = sin (2x) y = sin (x - /2) Does the **graph** ever **shift** vertically? – YES y = A sin (Bx – C)/**Graphing** Guidelines 1.Calculate period 2. Calculate interval (period / 4) 3.Calculate phase **shift**-‘start-value’ 4. Create X/Y chart w/ 5 **key** points from ‘start-value’ Equation of a Cosine Curve 0 x y 180 360 2 90 /2 270 3 /2 1 2 -2 y = A cos (Bx – C)**amplitude** = | A | period = 2 B phase **shift**/

c(i) + i 2 V w(i,j) Grouping & **amplitude** scaling is modeled as facility location Complete **graph**, every signal is represented by a node Cost opening a facility: /same group at time t Part of IBT dataset Similarity Query Compression by Interval Sharing **Key** Idea: If two sensors have near overlapping time series they can share a part/ humidity, sampling rate once every 30 seconds Signals not overlapping, but still correlated **Shifting** or scaling may help Question: Can we exploit this correlation ? We propose a/

**graph** reflects about the x-axis. c indicates the phase **shift**, also known as the horizontal **shift**. b affects the period. y = a tan b (x - c) y = a cot b (x - c) Unlike sine or cosine **graphs**, tangent and cotangent **graphs** have no maximum or minimum values. Their range is (-∞, ∞), so **amplitude**/ **shift** of a cotangent **graph**. For a cotangent **graph**, c is the value of x in the **key** vertical asymptote. For this **graph**, c = because the **key** asymptote **shifted** left to . An equation for this **graph** can be written as or **Graphs** /

(X) y 2 = sin (1/2x) Press **Graph** Y = A sin (Bx ± C) ± D **Amplitude** Period Change 2 |b| Vertical **Shift** up/down Horizontal **Shift** Left/Right C B Examples-- Y = 3sin (x + ) -3 A= _______ Period = _____ Horizontal Phase **Shift** = _________ Vertical Phase **Shift** = ________ Y = cos (x - 2 ) A= _______ Period = _____ Horizontal Phase **Shift** = _________ Vertical Phase **Shift** = ________ Y = -2sin (2x) A= _______ Period/

. Its x-intercepts are of the form x = n . Its period is . Its **graph** has no **amplitude**, since there are no minimum or maximum values. The **graph** is symmetric with respect to the origin, so the function is an odd function. For all x in the/ phase **shift** is units to the right. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 25 Example 5 **GRAPH** y = c + acot(x – d) (cont’d) To locate adjacent asymptotes, solve Divide the interval into four equal parts and evaluate the function at the three **key** x-/

= tan x, there is no phase **shift**. The y-intercept is located at the point (0,0). We will call that point, the **key** point. Next Back A tangent **graph** has a phase **shift** if the **key** point is **shifted** to the left or to the right. /Unlike sine or cosine **graphs**, tangent and cotangent **graphs** have no maximum or minimum values. Their range is (-∞, ∞), so **amplitude** is not defined. However, it is important to determine whether a is positive or negative. When a is negative, the tangent or cotangent **graph** will “flip” or /

x = nπ. Its period is π. Its **graph** has no **amplitude**, since there are no minimum or maximum values. The **graph** is symmetric with respect to the origin, so the /**shift** is units to the right. Copyright © 2009 Pearson Addison-Wesley1.1-25 4.3-25 Example 5 **GRAPHING** A COTANGENT FUNCTION WITH VERTICAL AND HORIZONTAL TRANSLATIONS (continued) To locate adjacent asymptotes, solve Divide the interval into four equal parts to obtain the **key** x-values Evaluate the function for the **key** x-values to obtain the **key**/

at (0,0), ( ,0) and (2 ,0). Parent Function **Key** Points * Notice that the **key** points of the **graph** separate the **graph** into 4 parts. y= a sin b(x-c)+d a = **amplitude**, the distance from the center to the maximum or minimum. If |a|/-c) + d d= vertical **shift** If d is positive, **graph** **shifts** up d units. If d is negative, **graph** **shifts** down d units. In trigonometric functions, these vertical **shifts** are called the vertical displacement. y = sin x +2 What changed? Which way did the **graph** **shift**? By how many units? y /

**Graph** transformations of the cosecant and secant functions Set your viewing window to the following settings: x min: 0 x max: 2π x scl: π/6 y min: -2 y max: 2 y scl: 1 Change your table to the following setting: Independent: 2. Ask **Amplitude**: a Period: π/b Phase **Shift**: c Vertical **Shift**: d **Key**/3π/4 π0 xcot x 0UD π/41 π/20 3π/4 πUD y = 2 tan x Page 196 #25 Page 196 #31 **Amplitude**: a Period: 2π/b Phase **Shift**: c Vertical **Shift**: d **Key** Points xcsc x 0UD π/21 πUD 3π/2 2πUD xsec x 01 π/2UD π 3π/2UD 2π1 Page 196 #37 /

Today you will use **shifts** and vertical stretches to **graph** and find the equations of sinusoidal functions. You will also learn the 5-point method for sketching **graphs**. What does the “normal” sine **graph** look like? What does the “normal” cosine **graph** look like? Sine Wave Tracer Draw the “normal” **graph** for each. Label the 5 **key** points. Maximum? Minimum? Crossings? Period? **Amplitude**? Parent Equation General Equation Period: 2/

o and 2 , there are two zeros at Parent Function **Key** Points * Notice that the **key** points of the **graph** separate the **graph** into 4 parts. y = a cos b(x-c)+d a = **amplitude**, the distance between the center of the **graph** and the maximum or minimum point. If |a| > 1/ c is positive, the **graph** **shifts** right c units (x-c)=(x-)+c)) What changed? Which way did the **graph** **shift**? By how many units? y = a cos b(x-c) + d d= vertical **shift** If d is positive, **graph** **shifts** up d units If d is negative, **graph** **shifts** down d units y /

at drawing these **graphs** using **graph** paper is strongly recommended. Tangent and Cotangent Three **key** elements of tangent and cotangent: 1.For which angles are tangent and cotangent equal to 0? These will be x-intercepts for your **graph**. 2.For /**graph** between the asymptotes. y = tan x y = cot x Transformations |A| = **amplitude** π/B = period (distance between asymptotes). C/B gives phase **shift** from zero. Examples—**Graph** the Following Secant and Cosecant The **graphs** of secant and cosecant are derived from the **graphs**/

and use even-odd properties to simplify 1)Find **Amplitude** and period 2)Find Phase **Shift**, and vertical **shift** 3)Find starting and ending x-coordinates 4)Divide into 4 equal parts 5)Label **key** points 6) Connect Remember, cos(x) = cos/**Amplitude** = T = P.S. = V.S. = Starting point is phase **shift**. Ending point is Phase **shift** + Period 3 5 + You will take the starting and ending points and find the average, then find the average again to break it up into four equal regions You want to study the sine and cosine **graphs**/

two **key** points on the cosecant **graph** are. In the remaining examples we will use a sine or cosine **graph** as an aid in sketching the **graph** of a cosecant or secant function, respectively. cont’d 20 Example 7 **Graph** one cycle of Solution: First, we sketch the **graph** of There is a vertical translation of the **graph** of y = sin x one unit downward. The **amplitude** is/

the new end (bx - c = π) 4.Find the new interval (new period / 2) to divide the new reference period into 2 equal parts to create new x values for the **key** points 5.Adjust the y values of the **key** points by applying the **amplitude** (a) and the vertical **shift** (d) 6.**Graph** **key** points and connect the dots

, with a roll-off factor a=0.5. The carrier frequency is f c =40/T. evaluate and **graph** the spectrum of baseband signal and the spectrum of the **amplitude**-modulated signal Carrier-Phase Modulation This type of digital phase modulation is called Phase-**Shift**-**Key** where gT(t) is the transmitting filter pulse shape. when gT(t) is a rectangular pulse we expressed/

Sine and Cosine Functions Section 6 Objectives **Graph** transformations of the sine function **Graph** transformations of the cosine function Determine the **amplitude** & period **Graph** using **key** points Find an equation for a sinusoidal **graph** **Amplitude**: a Period: 2π/b Phase **Shift**: c Vertical **Shift**: d **Key** Points xsin x 00 π/21 π0 3π/2 2π0 xcos x 01 π/20 π 3π/20 2π1 Page 187 #25 Page 187 #33/

Section 4-5 2 Objectives I can determine all **key** values for 6 trig functions I can **graph** cosine functions I can determine **amplitude**, period, and phase **shifts** of cosine functions 3 Cosine Function **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/

asymptotes 2.Find T 3.Find 3 **key** points **Graph** y = 2 cot x –2 **Graph** y = cot(4x – π/2) 0 < Bx - C < π **Graphing** Trigonometric Functions Standard form: y = A tan (Bx - C) + D or y = A cot (Bx - C) + D **Amplitude** = “none” (but it does // B, (C + π) / B Vertical **shift** = D Domain (tan x): -∞ ≤ x ≤ +∞, x ≠ n(π/2), n odd integer Domain (cot x): -∞ ≤ x ≤ +∞, x ≠ n(π) Range: -∞ ≤ y ≤ +∞ Three **key** points: (, ), (, ), (, ) Drawing the curve: (1) Determine D, the vertical phase **shift**. This will be the “new” horizontal axis/

Logarithmic Form Example 3: Evaluate Logarithmic Expressions **Key** Concept: Parent Function of Logarithmic Functions Example 4: **Graph** Logarithmic Functions **Key** Concept: Transformations of Logarithmic Functions Example 5: **Graph** Logarithmic Functions Example 6: Real-World Example/ Standards F.IF.7.e **Graph** exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and **amplitude**. F.BF.3 Identify the effect on the **graph** of replacing f(x) by/

**graph**. The points are collected over a number of samples to produce a scatter plot. Data **Graph** Spectrum Analyzer Probing This probe can be used as a multi-channel **graph**, displaying **amplitude**/value in a column. **Shift** The input data is adjusted to either move the inputs back by a specified **shift** value to do predictions /learning environment Emphasis on understanding rather than focusing on complex mathematical derivations **Key** concepts reinforced by interactive examples Over 200 fully functional simulations of /

divisible into shorter cycles of similar character with **amplitudes** approximating their own.” -- Burns and Mitchell, 1946 Recessions /economic growth Recession years are shaded blue: note downward slope on **graph** indicating that GDP is decreasing. U.S. real gross domestic/2002 by The McGraw-Hill Companies, Inc. All rights reserved. Three **Key** Aspects of the Economy Economic Growth Inflation Employment The State of the / plunge in the demand for hides Thus small **shifts** in demand growth at the consumer level are amplified/

School District in Alabama to Receive Accreditation – **Shifts** Accreditation Burden from Schools to the School District – Avoids $170,000 in Accreditation Costs 2012 – 2013 **Key** Accomplishments District-Wide Talent Management & Development District/ between logarithmic & exponential equations; **graph** each Use law of cosines & law of sines **Graph** & determine domain, range, **amplitude** & period of cosine & sine functions Number Sense, Quadratic Functions, and Matrices **Graph** & solve quadratic equations & inequalities/

a vertical reflection and a horizontal **shift**. By remembering a few **key** aspects of the **graphs** (their patterns), you can easily and quickly sketch the **graphs** of tangent and cotangent functions. Tangent vs. Cotangent – **Key** Components At x=0:- The /**graph** has the “other” characteristic. The tangent goes up (left to right), the cotangent goes down. Half way in between the primary components, the curve has a point the same distance as the **amplitude** up (or down) from the axis. ASSIGNMENT Unit 4 Assignment 2 – **Graphs**/

2 3 y = sin x, 0 < x < 2 The **Graph** of y=sinx **Graphing** Variations of y=sinx Identify the **amplitude** and the period. Find the values of x for the five **key** points – the three x- intercepts, the maximum point, and the minimum point. Start/**graph** of y = A sin Bx has **amplitude** = | A| period = 2 /B. The **graph** of y = A sin Bx has **amplitude** = | A| period = 2 /B. x y y = A sin Bx **Amplitude**: | A| Period: 2 /B **Amplitudes** and Periods The **graph** of y = A sin (Bx – C) is obtained by horizontally **shifting** the **graph**/

interval 15 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 3 **Graphing** y = sin bx Solution continued Divide this interval into four equal parts to find the x-coordinates for the **key** points: (y values do not change.) To find the period of divide/). 24 © 2010 Pearson Education, Inc. All rights reserved PROCEDURE FOR **GRAPHING** y = a sin b(x – c) AND y = a cos b(x – c) Step 1Find the **amplitude**, period, and phase **shift**. **amplitude** = |a| phase **shift** = c period = Step 2The starting point for the cycle is x/

impart information on the optical carrier. **Amplitude**, E 0 [1]**Amplitude**, E 0 ; the format that modulates the **amplitude** of the optical carrier is called “**amplitude** modulation”. If the information is digital then the format is known as “ **amplitude** **shift** **keying**” or ASK for short. The/ the receiver in normalise form. The ratio R/B represents the spectral efficiency whose upper limits is C/B. The **graph** in the next slide illustrates the Harley – Shannon law. The curve corresponding to R = C separates the regions; /

identify transformations and state how the **key** features have changed (iii) **Graph** and analyze y = tan( ½ x + /4) – 3 identify transformations and state how the **key** features have changed 10/9/201537 Math 2 Honors - Santowski 38 (G) Writing Sinusoidal Equations ex 1. Given the equation y = 2sin3(x - 60 ) + 1, determine the new **amplitude**, period, phase **shift** and equation of the axis of the/

go back to the quadratic equation in **graphing** form: y = a(x – h) 2 + k If a < 0: reflection across the x axis |a| > 1: stretch; and |a| < 1: shrink (h, k) was the vertex (locator point) h gave us the horizontal **shift** k gave us the vertical **shift** Transforming the Cosine (or Sine (Sinusoid) **Key** Points **Amplitude** – increase the output by a factor of/

Functions Use reciprocal identities **Graph** of y = sec x **Key** Points **Graphing** the Tangent Function Properties of the Tangent Function Transformations of the **Graph** of the Tangent Functions **Graphing** the Cotangent Function **Graphing** the Cosecant and Secant Functions Phase **Shifts**; Sinusoidal Curve Fitting Section 5.6 **Graphing** Sinusoidal Functions y = A sin(!x), ! > 0 **Amplitude** jAj Period y = A sin(!x { Á) Phase **shift** Phase **shift** indicates amount of **shift** To right if/

FSK, and PSK. **Amplitude**-**Shift** **Keying** 2 binary values represented by 2 **amplitudes**. Typically, “0” represented by absence of carrier and “1” by presence of carrier. Prone to errors caused by **amplitude** changes. Frequency-**Shift** **Keying** 2 binary values represented by/ AS backbone; all areas connected to it. OSPF 3 Type of service routing: Routing updates: Uses different **graphs** labeled with different metrics. Routing updates: Adjacent routers exchange routing information. Adjacent routers are on different LANs. /

Techniques: ASK, FSK, and PSK. Techniques: ASK, FSK, and PSK. 60 **Amplitude**-**Shift** **Keying** 2 binary values represented by 2 **amplitudes**. 2 binary values represented by 2 **amplitudes**. Typically, “0” represented by absence of carrier and “1” by presence of /; all areas connected to it. 376 OSPF 3 Type of service routing: Type of service routing: –Uses different **graphs** labeled with different metrics. Routing updates: Routing updates: –Adjacent routers exchange routing information. –Adjacent routers are on different/

where sin(t) >0 sin(t) > 0 The Sine Function Sine Function **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./**amplitude**, period, and phase **shift** of y = 2sin(3x- ) Solution: **Amplitude** = |A| = 2 period = 2 /B = 2 /3 phase **shift** = -C/B = /3 Example cont. y = 2sin(3x- ) **Amplitude** Period: 2π/b Phase **Shift**: -c/b Vertical **Shift** 6.6 More Trig. **Graphs** Objective: Students will look at more trig. **graphs**/

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