Graphing trig functions with transformations worksheet
Graphing trig functions with transformations worksheet. 25 mins •website links • Practice graphing sine and cosine worksheet • Cumulative Bellwork notebook • Online graphing calculator desmos. This lesson allows the students to investigate the various transformations for themselves using an online graphing software before combining the rules to solve exam-style questions on graph transformations. Write down the coordinates of the point A. m Worksheet by Kuta Software LLC Answers to Graphing Tangent Feb 28, 2024 · Transformation function graphs horizontal vertical rules stretches transformations graph examples translations compressions functions stretch down tables solutions worksheets onlinemathlearning theirWorksheet transformations graph function quadratic functions worksheeto via Function transformation worksheet50 graphing trig functions worksheet. transformationof . docx, 74. If there is a period change, find the new intervals first, then graph the parent graph as usual. Equation (6) = _____ (in terms of the sine function) Graph one complete period of the given sine or cosine curve. 1) y = sin (θ − 135) Feb 20, 2019 · Trigonometric Graphs and their transformations. As we know, the graph of ( ) 2 3. This follows chapter 5 of the grade 11 Functions McGraw Hill textbook and chapter 6 of the grade 11 Functions Nelson textbook. Example: 2 sin (2x - 4) + 3; C = 4. Use the information you gathered in 10 to write an equation for the graph. Level up on all the skills in this unit and collect up to 1,700 Mastery points! Let's extend trigonometric ratios sine, cosine, and tangent into functions that are defined for all real numbers. There are two types of transformation: translations and reflections, giving 4 key skills you must be familiar with. (Total for question 2 is 4 marks) x y O (2, –3) y = f(x) The coordinates of the minimum point of this curve are (2, –3). U6D7_W_Trigonometric Applications: 8 : Solving Trig Equations. The reader should be familiar with the trig ratios, using radians and working with exact values which arise from the following standard triangles. PHASE SHIFT. Knowledge application - use We can graph \(y=\csc x\) by observing the graph of the sine function because these two functions are reciprocals of one another. Plus each one comes with an answer key. on Practice graphing sine and cosine worksheet. Feb 22, 2016 · This trigonometry and precalculus video tutorial shows you how to graph trigonometric functions such as sine and cosine functions using transformations, phas Example 3: applying a reflection in the x- axis. Graphing Tangent Transformations Graph each function using radians. The graph crosses the axis at \mathbf{90}\degree and \mathbf{270}\degree. 21) y = 4 cos (πx) Mathster is a fantastic resource for creating online and paper-based assessments and homeworks. Worksheet by Kuta Software LLC Precalculus Graphing Trig Functions Practice Name_____ Date_____ Period____ ©V P2X0v1t6t jKSuctral wSBocfYtvw[aGrUeu uL[LzCu. Trig, Sin, Sine, Cos, Cosine, Tan, Tangent Unit test. Lesson finished with an interactive plenary where students need to evaluate trigonometric values. 3 and 3. -1-Find the amplitude, the period in radians, the minimum and maximum values, and two vertical Worksheet by Kuta Software LLC Kuta Software - Infinite Precalculus Graphs of Trig Functions Name_____ Date_____ Period____-1-Find the amplitude, the period in radians, the phase shift in radians, the vertical shift, and the minimum and maximum values. (This sheet is a summative worksheet that focuses on deciding when to use the law of sines or cosines as well as on using both formulas to solve for a single triangle's side or angle) Law of Sines. 2 TRANSFORMATIONS OF SINE AND COSINE GRAPHS To obtain the graph of y a sin b(x h) k or y a cos b(x h) k, transform the graph of y a sin bx or y a cos bx as follows. 5 is x = 210°. Resource type: Worksheet/Activity. Unit 1 Right triangles & trigonometry. Suitable for all the big examination boards including AQA, Edexcel, OCR and WJEC. _____ 3] Phase shifting a tangent function changes its domain when compared to the Apr 22, 2023 · This is a set of worksheets aimed at higher level students looking at the transformations of graphs. 1) y = −sin x + 1 2) y = −3 cos 2x 3) y = §· ¨¸ ©¹ 4csc x 2 4) y = 2 tan 4x The graph of f(x) = x2 is horizontally stretched by a factor of 3, then shifted to the left 4 units and down 3 units. Two cycles of the graph are shown to illustrate the. 1-5 Assignment - Parent Functions and Transformations. One solution of sin x = −0. MCR3UI U6D7_Trigonometric-Applications VIDEO: Pg. Unit 2 Trigonometric functions. Microsoft Word - Trig Graphs Worksheet. g(x) = 5(x + 3)2 − 2. For b > 0, the period of y = a sin bx is 2 π . 1) y sin ©z 82g0 H142C 0KCu7tRaF ySBoJfMt7wwafr Ies HL3L DCt. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance Explore math with our beautiful, free online graphing calculator. _____ 2] Vertically displacing a cosecant function up 2 units changes its range compared to the cosecant parent graph. Determine whether the transformation is a translation or reflection. Mar 14, 2023 · The sine and cosine functions have several distinct characteristics: They are periodic functions with a period of 2π. tan O= 0 Graph each function on the given interval II. Probability using permutations and combinations. Graphs of Trigonometric Functions Videos 338, 339 & 340 on www. Printable in convenient PDF format. reflection in the y-axis. 5-a-day GCSE 9-1; 5-a-day Primary; 5-a Explore math with our beautiful, free online graphing calculator. Cazoom also has created graphing worksheets with ways for your child to interact with transformations of common trigonometric graphs. Then sketch a graph of the transformation. As with the sine graph, this portion is one period of the graph, so it is repeated for all the values before 0\degree and past 360\degree. Topics in this unit include: periodic functions, graphs of sine and cosine, transformations of trig functions, and applications of trig functions. . Feb 13, 2013 · 2. We can think graphs of absolute value and quadratic functions as transformations of the parent functions |x| and x². This topic is about the effects that changing a function has on its graph. pptx, 10. Graph the function. There are 5 worksheets covering translation, stretching, reflection, sketching curves and trigonometric transformations. Approximating area under a curve. graph the sine function (in degrees and radians ), graph the cosine function (in degrees and radians ), recognize the periodicity of the sine and cosine functions and other key properties such as maximum and minimum value (and range) and their roots, graph simple transformations of sine and cosine, for We begin with worksheets with the properties of sine, cosine, and tangent measurements and then move to trigonometric word problems (both with and without pictures). ( π 2 x) − 4 in the interactive widget. = A * sin [ (B * x - C)] + D. yt =sin( ) and then connect Feb 22, 2018 · Trig graph transformations worksheet. Explore the relationship between these functions and the unit circle. Such behavior is referred to as periodic. Resource type: Lesson (complete) File previews. The domain of each function is ( − ∞, ∞) and the range is [ − 1, 1]. tan (-180 °) 3. Find another solution of sin x = −0. Fill in the blanks to complete the table. T w tAblulm ZrciVgKhMtOsb Vrmeis^eRrGvFeDdS. This is the fourth lesson that I would teach on trig graphs and builds on an understanding of transforming functions, by looking at Transforming Trigonometric Graphs. Click here for Answers. parallel to the x-axis. 3. doc Worksheets are Transforming trigonometric graphs, Transformation on trigonometric functions, Graphs of trigonometric functions, Work 15 key, Graphs of trig functions, Trigonometry work t1 labelling triangles, Graph transformations, Chapter 45 part 1 trigonometry in radians. C = horizontal shift. P (2,5). Day 2 April 24, 2020. Math. Identify period and phase shift - and amplitude if it applies. n O BAElVly nrriDgGhVtKsA LrceasKeHrzvSe\d\. 5 y = cos(x) Sketching trig functions. Here is the graph of y = sin x for 0 ≤ x ≤ 360. Use your graph to solve cos x = 0. csc (720 °) 6. Note that one moveable point always defines an extremum point in the graph and the other point always defines a neighbouring intersection with the midline. cos (540 °) 4. This worksheet is aimed at learners undertaking a Further Maths GCSE course and is also suitable for A-level maths students, however, many of the questions can be used to extend the skills of Higher tier GCSE students. 2 sin (2x + 4) + 3; C = -4. GRAPHING. The test will help you with these skills: Making connections - use understanding of sine and cosine transformations to identify which graph represents a given function. Test your knowledge of the skills in this course. find the coordinates of a point on a trigonometric graph after it has been transformed, translate the graph of a trigonometric function in the directions of the 𝑥 - a x i s and 𝑦 - a x i s, stretch the graph of a trigonometric function. A transformation is something that is done to a graph/function that causes it to change in some way. U3D2_T Graphing the Reciprocal and Root Functions : Master graphing the “root function”, the “reciprocal function” (and the asymptotes), the Writing functions cont’d Quiz - Graphing Sine and Cosine Functions Worksheets pp. 1) y = sin (θ − 135) Dec 1, 2021 · This video shows how to transform the trigonometric graphs for GCSE Maths. Graph the sinusoidal function. Trigonometry: All the Trig Functions | Desmos Graphing Tangent Transformations Graph each function using radians. g(x) = 4(x + 1)2 − 5. 4 Sine and Cosine Transformations Worksheet Determine the amplitude, period, frequency, phase shift, and vertical translation for each. Find the amplitude, the period in radians, the phase shift in radians, the vertical shift, and the minimum and maximum values. Graph Transformations. Central Focus. 70. ©z 82g0 H142C 0KCu7tRaF ySBoJfMt7wwafr Ies HL3L DCt. 1) For the pair of functions . zip, 3. Power rule for differentiation. Combinations. Vertical shiftShift the graph k units vertically. Over 100 individual topics extend skills from Algebra 2 and introduce Calculus. Thus the y-coordinate of the graph, which was previously sin (x) , is now sin (x) + 2 . 2, 3. This worksheet starts with some notes on Trig graphs (written very hastily so Word version attached for editing) and then gradually introduces each type of transformation with matching 1. Google Classroom. Transformations Of Graphs 1 - Translation - given two curves state the translation vector and the equation of the second graph. Explore math with our beautiful, free online graphing calculator. n 7 LMla ud xeU ywli Ptohs 0IwnLf2iwnai StXen yAhlQggeGbQrXa7 H2q. Students will practice: a) identifying period and amplitude based on equation or on the graph, b) write equation from graph, c) write graph from equation. 5 mins : 10 mins . Graphs of Trig Functions . 1 – Inverse Trig Functions . Then sketch the graph using radians. Math 165 – Section 7. U6D8_S_SolvingTrigEquations. a cos bx is also 2 π . 2 4 6 8 − 4 − 6 − 8 2 4 6 8 − 4 − 6 − 8. docx, 811. The blue curve represents the function \ (y=\cos (2x)\) and is compressed horizontally by a factor of 2. U6D8_T_SolvingTrigEquations Graph the image of the figure using the transformation given. yt=sin( ) is given below with six key points emphasized. Default Trig Function Graphs: Knowing the 4 Segments of Trig Transformations: = amplitude. 42 MB. • Period: the number of degrees or radians we must graph before it begins again. Graphs of the Trigonometric Functions Find each value by referring to the graphs of the trigonometric functions. Independent and dependent events. -! -0. y = −f (x). tan (180 °) 5. Analyze the equation to determine the features of the graph of each function. 69. 4. You will also need to come up with the equation from the graph. -1-Using degrees, find the amplitude and period of each function. 5. π. davenportmath. 52 KB. Transformations of periodic functions. Worksheet with axes drawn on for students to produce the various different transformations (translation / reflection / stretch) of the sinx curve. Unit 4 Trigonometric equations and identities. D = vertical shift. Ambiguous Case of the Law of Sines. The graph of sine is shown as a dashed orange wave so we can see the relationship. *Click on Open button to open and print to worksheet. 7. 1-5 Bell Work - Parent Functions and Transformations. Level up on all the skills in this unit and collect up to 1,900 Mastery points! Discover how to measure angles, distances, and heights using trigonometric ratios and the unit circle. Do not use a graphing calculator . Here are some examples of drawing transformed trig graphs, first with the sin function, and then the cos (the rest of the trig functions will be addressed later). 5 for 0 ≤ x ≤ 360. Free Algebra 2 worksheets created with Infinite Algebra 2. Law Of Cosines. The graph of the tangent function combines the periodic behavior of a trig function (with period π) along with the algebraic SOLUTIONS: Week 3 Practice Worksheet . Use your results from above to nd the amplitude and period for the following functions. This will be the last time we graph two cycles to illustrate the. File previews. 17 # 4 – 9 (show work on separate paper) p. 5! 0 0. sin 0 =-1 8. Learn how to graph the sine and cosine functions using their amplitude, period, phase shift, and vertical shift. The lesson will launch by recognizing sinusoidal waves in the real world such as visible Apr 4, 2018 · The Corbettmaths Practice Questions on Trig Graphs. If we repeat this period a few times, we will see that the shape is exactly the same as the sine graph. Before, getting on to solving trigonometric equations using the graphs. Law of Sines and Cosines Worksheet. Graph sinusoidal functions. 71. Key points. + 3π), identify (i) the amplitude, (ii) the phase shift, and 20) (iii) the period. Instantaneous rates of change. Suitable for GCSE Students sitting the 9 – 1, A-Level Maths students revision and some Key Stage 3 students. Find the amplitude, the period in radians, and the phase shift in radians. Trigonometry - graphs of functions. Start Course challenge. f x x and f x x( ) sin=−−11. Learn how to use sine, cosine, and tangent to solve real-world problems involving triangles and circular motion. Each function has a domain (−∞, ∞) and a range [−1, 1]. Step #2: Write the transformations IN THE ORDER IN WHICH THEY OCCUR Step #3: Graph each transformation – one at a time, use more than one J D xMSad7eH kw Ti atrh6 vI rnAfqiEnciYtje 5 PAtl Ygae xb Lrwat E2 Z. Students will learn to recognize the graphs which will be important in future lessons as well as in other disciplines. The types of function transformation included are:- vertical stretch, horizontal stretch - vertical and horizontal shift- reflections about x-axis and y-axisThe first page also includes a brief review of function transformation. You will probably be asked to sketch one complete cycle for each graph, label significant points, and list the Domain , Range , Period and Amplitude for each graph. Verify you are correct by graphing each. 09 KB. Area under a curve by limit of sums. com Question 12: Here is the graph of for One solution of is Find another solution of Question 13: Here is the graph of (a) Use the graph to solve (b) Use the graph to solve y = sin(x) 0 ≤ x ≤ 360 sinx = −0. n Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Translating Graphs of Trig Functions Date_____ Period____ Using degrees, find the amplitude and period of each function. Step #2: Write the transformations IN THE ORDER IN WHICH THEY OCCUR Step #3: Graph each transformation – one at a time, use more than one Sep 18, 2018 · U3D1 Worksheet Solutions Functions Relations D & R: 2: 3. Find the period and amplitude of the sine function below. sin (-720 °) 2. Trig applications. Identify the features of the graph, then sketch the graph NEATLY Unit test. Learners will need to sketch the graphs to describe the transformations of some trigonometric functions in different intervals Infinite Precalculus covers all typical Precalculus material and more: trigonometric functions, equations, and identities; parametric equations; polar coordinates; vectors; limits; and more. Unit test. b. The graph of . 6 Properties of Trigonometric functions Section 1 Review of Trigonometry This section reviews some of the material covered in Worksheets 2. 2 The graph of y = f(x) is shown below. y t = −sin. Example 1: Sketch the graph of y = 3 + sin 2x Introduction to Trigonometric Identities and Equations; 7. Verify your answer with Desmos. Write down the coordinates of the point B. 31 KB. pGraph translations and reflections of tangent graphs. 2: Graphing and finding properties of the root function and the reciprocal function. This is a whole lesson on learning how to Transform Trig Graphs and draw or name the transformation. Permutations vs combinations. Click here for Questions. 4 Sum-to-Product and Product-to-Sum Formulas; 7. 63 MB. 1 Solving Trigonometric Equations with Identities; 7. Step #1: Start by graphing the parent function = if there is no period change (b). 3] _____ Changing the phase shift of a tangent function changes its domain. com . 1 (below) # 3 – 6 Tuesday 10/29 Application problems cont’d Quiz – Writing sine Students will be practice working equations of sine, cosine, tangent periodic graphs with phase shifts, change in amplitude and period. Motion along a line. is a . Sample spaces and The Counting Principle. yt =sin( ). 3 Double-Angle, Half-Angle, and Reduction Formulas; 7. Objective Learning Activity Duration Materials needed Sine and Cosine graphs with Transformations Worksheets are Transformation on trigonometric functions, Trig transformations, Transforming trigonometric graphs, Chapter 5 trig functions, Trigonometric formula de nition of the trig functions, Chapter 5 trig functions, Trigonometry word problems, Graphs of trig functions. 6 Modeling with Trigonometric Functions www. Subject: Mathematics. The period for sin x and cos x is 2π. They have kindly allowed me to create 3 editable versions of each worksheet, complete with answers. Jun 4, 2020 · Lesson starts with plotting the graphs of Sine, Cosine and Tangent functions. Parent Functions and Transformations Worksheet, Word Docs, & PowerPoints. This was taken from my 150 Page workbook which covers many more maths topics. 1. This lesson shows how to graph the reciprocal trigonometric functions (y = csc x, y = sec x and y = cot x) using the y = sin x, y = cos x and y = tan x functions. 4] _____ Changing the vertical stretch of a cotangent function changes its domain. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more. 2 Sum and Difference Identities; 7. All values of y shift by two. Welcome; Videos and Worksheets; Primary; 5-a-day. Phase shift is any change that occurs in the phase of one quantity, or in the phase Worksheet by Kuta Software LLC Algebra 2 Graphing Trig Functions Phase and Vertical Shifts Name_____ ID: 1 Date_____ Period____ ©q y2r0p1q5d KKJuUtAaR XSTo[fWtQwNatrGed `LYLzCm. Then brings in sketching a comparison of the Sine and Cosine functions. Age range: 14-16. The transformation y = f(ax) is a horizontal stretch of y = f(x) with scale factor 1. Sine and Cosine Graph Transformations quiz for 8th grade students. 6. The main worksheet for this lesson has Oct 11, 2023 · Age range: 16+. Where the graph of the sine function decreases, the graph of the cosecant function increases. This is a full lesson that I’ve made on graph transformations. sec _ =-1 9. 2 ˇ 4 ˇ 4 1 1 p ˇ 3 ˇ 6 1 p 3 2 Learn how to graph trigonometric functions with Kuta Software, a powerful and easy-to-use math software for teachers and students. Sketch the graph and state the coordinate of the image of point P P on the graph y=-f (x). MCR3UI_U6D7_Warmup_Identifying Equations VIDEO. Free Precalculus worksheets created with Infinite Precalculus. See Figure \(\PageIndex{7}\). The graph of y = sin x is symmetric about the origin, because it is an odd function. Each question also includes a graph of either sin(x) or cos To shift such a graph vertically, one needs only to change the function to f (x) = sin (x) + c , where c is some constant. There will be a word problem where you will be given a situation and must come up with the 1 . Test and worksheet generator for Precalculus. 5! ! 1. Could easily be adapted to cosx / tanx also. Then graph. 2. p Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Graphing Trig Functions Date_____ Period____ J D xMSad7eH kw Ti atrh6 vI rnAfqiEnciYtje 5 PAtl Ygae xb Lrwat E2 Z. The green curve represents the function \ (y=3\cos (2x)\) and is the blue curve stretched vertically Sep 18, 2018 · U6D7_T_Trigonometric Applications. 5 Solving Trigonometric Equations; 7. This will include a graph and must come up with your own word problem that corresponds to the given graph. Microsoft Teams. Identify function transformations. U3D2_S Basic Grade 11 Graphs. Goals pGraph translations and reflections of sine and cosine graphs. Find other quizzes for Mathematics and more on Quizizz for free! Worksheet 4. B M EA wl5lD 2r Yilg Shdt esv RrJeLsye2r7vge xd g. The diagram shows the graph of y=f (x) y = f (x) and a point on the graph P (2,5). Use what we know about graph transformations to “transform” the six key points on . 2 ˇ 4 ˇ 4 1 1 p ˇ 3 ˇ 6 1 p 3 2 Graphing Sine and Cosine Trig Functions WS #3. Section 4. Part 1: See what a vertical translation, horizontal translation, and a reflection behaves in three separate examples. Permutations. In this lesson students will explore how to graph trigonometric graphs and identify the period, amplitude, and asymptotes. 5 x = 330 sinx = −0. g is a transformation of f . For the following exercises, describe how the formula is a transformation of a toolkit function. The worksheet increases in difficulty level as it progresses. Here is a graph of y = cos x for 0 ≤ x ≤ 360. = period. Indefinite integrals. Note: If C is +, the shift will be negative, aka to the left. Importantly, we can extend this idea to include transformations of any function whatsoever! This fascinating concept allows us to graph many other types of functions, like square/cube root, exponential and Definition of the derivative. About this unit. phase shift. 1-5 Guided Notes SE - Parent Functions and Transformations. The graph below shows f as a solid blue line and g as a dotted red line. Students will be able to. What is the formula of g in terms of f ? Worksheet 4. Lesson Plan. Worksheet Name. Both exhibit a repeatable pattern that is common to all trig functions. Again, we graph two cycles to illustrate the. The period of a function is the x interval needed for the function to complete one cycle. 14 – 15 Monday 10/28 Application problems (Notes page 1 # 1 – 2: see below; And page 16 # 1 – 3) Word problem worksheet p. OpenStax offers free online textbooks and resources for precalculus and other math courses. 11. sec (180 °) Find the values of _ for which each equation is true. Then graph the function. 2sin 1 2 cos 1 2 12. U3D2_S Warmup D & R. 388: #12a – d (assume t = 0 is low tide), #14, #16 & Worksheet: Trig Graphing Applications. Label your graphs with correct units on the x- and y- axis. Trigonometry 4 units · 36 skills. 5! 2!-3 Feb 13, 2024 · Graphs trigonometric functions sec sine worksheet cosine graphing practice chapterGraphs trigonometric trig formulas maths trigonometry equations graphing derivative calculator tutor shelovesmath Worksheet trigonometry trigonometric functions graph reviewed curatedFunctions trigonometric graphing graph period amplitude act where. For b > 0, the period of y =. y = 3cos. Mutualy exclusive events. 1. corbettmaths. Importantly, we can extend this idea to include transformations of any function whatsoever! This fascinating concept allows us to graph many other types of functions, like square/cube root, exponential and Example: Sketch the functions \ (y=\cos (x), y=\cos (2x),\) and \ (y=3\cos (2x)\). Graph each of the following sine or cosine transformations over ONE PERIOD. 1-5 Exit Quiz - Parent Functions and Transformations. Date: Per: y:2sin3x y : 3 cos 4x y : -cos 2x — 5 sin2x F L y -sin (x —T) HT-. Sep 18, 2017 · Trigonometric Functions. Take the reciprocal of each value and plot the ordered pair in the coordinate plane. Properties of Inverse Functions When graphing reciprocal trigonometric functions, first find the values of the original trig function. Describe the transformations required to obtain the trig function starting from the parent function. 14. Unit 3 Non-right triangles & trigonometry. Graph trig functions (sine, cosine, and tangent) with all of the transformations The videos explained how to the amplitude and period changes and what numbers in the equations. È y: 3 sin 6x—3 Jan 21, 2021 · A GCSE Maths Worksheet covering the transformation of trigonometric functions. Solution: The function\ (y=\cos (x)\) is shown in black. Write down the coordinates of the turning point of the curve with equation Then graph the function. The transformation y = f(–ax) is a horizontal stretch of y = f(x) with scale factor 1 parallel to the x-axis and then a. You might be surprised at how we can use the behavior of those functions to model real-world situations involving carnival rides and Graphing All Trig Functions Review Worksheet (Sin, Cos, Tan, Csc, Sec, & Cot) Sketch the graph of each function for one period. Course challenge. p Worksheet by Kuta Software LLC Kuta Software - Infinite Algebra 2 Name_____ Graphing Trig Functions Date_____ Period____ About this unit. XABlkgHe3btr taq f2 g. Determine the amplitude or vertical stretch, period, phase shift, and vertical shift for each. 1) translation: 5 units right and 1 unit up x y B G T 2) translation: 1 unit left and 2 units up x y M Y G 3) translation: 3 units down x y U Q L 4) translation: 5 units right and 2 units up x y I X E 5) translation: 4 units right and 4 units down x y A J I 6) translation: 2 units Mar 21, 2021 · docx, 336. m Worksheet by Kuta Software LLC Answers to Graphing Tangent Always (A), Sometimes (S), or Never (N) _____ 1] Changing the period of a sine function changes its domain compared to the sine parent graph. ot bx it dm sp qc iq em ii jb