MYP5E - Transformations of Trigonometric Functions

Transformations of Trigonometric Functions Assessment Type Mathematical Investigations Recommended Grade Level Grade 10 (MYP5) Extended MYP Assessment Criteria Criterion B: Investigating patterns Criterion C: Communicating MYP Criterion Level MYP 5 MYP Command Terms Used use, sketch, describe, state, select, write down, suggest, apply, find, verify, plot, explore, justify, present MYP Global Context Identities and relationships MYP Key Concepts MYP Related Concepts Relationships, Systems Representations, Models MYP Branch of Mathematics Thinking with models and Spatial reasoning • • • • MYP Topics and Skills Graphing trigonometric functions Transforming trigonometric functions Trigonometric ratios in right-angled triangles Radians • • • Prior Knowledge Needed Geometric transformations: translation, reflection, dilation Exact values of certain trigonometric functions (sine and cosine) Unit circles and exact values of sine/cosine of multiples of 30/45 degree angles Assessment Description In this assessment, students work through scenarios of changing the function rule of the sine and cosine functions by addition and/or multiplication and see how such change affects their graph. Students explore the graphs of • 𝑦 = sin(𝑥) + 𝐷 and 𝑦 = cos(𝑥) + 𝐷, • 𝑦 = 𝐴 sin(𝑥) and 𝑦 = 𝐴 cos(𝑥), • 𝑦 = sin(𝑥 − 𝐶) and 𝑦 = cos(𝑥 − 𝐶), and • 𝑦 = sin(𝐵𝑥) and 𝑦 = cos(𝐵𝑥). Students explore patterns, describe rules, and verify and justify their findings. Materials Needed A Graphic Display Calculator (GDC) is required. Task-specific instructions / Recommendations The use of pencil is recommended for sketching graphs. © 7on7 Academy 2023 Assessment Criterion B: Investigating patterns Achievement Level Descriptor (MYP5) The student does not reach a standard described by any of the descriptors below. The student is able to: The student is able to: i. apply, with teacher support, mathematical i. use their GDC to sketch the required graphs (Q1) problem-solving techniques to discover and briefly describe what they notice about these simple patterns graphs (Q2) ii. state predictions consistent with patterns ii. state their predictions about the relationships iii. (not demonstrated at this level). between the graph of 𝑦 = sin(𝑥) / 𝑦 = cos(𝑥) and the graph of 𝑦 = sin(𝑥) + 𝐷 / 𝑦 = cos(𝑥) + 𝐷 (Q3), (Q4) and sketch the required graphs (Q5) iii. (not demonstrated at this level). The student is able to: i. apply mathematical problem-solving techniques to discover simple patterns ii. suggest general rules consistent with findings iii. (not demonstrated at this level). The student is able to: i. use their GDC to sketch the required graphs in (Q6), (Q8) and briefly describe what they notice about these graphs (Q7), (Q9) ii. suggest a general rule about the relationship between the graph of 𝑦 = sin(𝑥) / 𝑦 = cos(𝑥) and the graph of 𝑦 = 𝐴 sin(𝑥) / 𝑦 = 𝐴 cos(𝑥), for both 𝐴 > 0 and 𝐴 < 0 (Q10) iii. (not demonstrated at this level). The student is able to: i. select and apply mathematical problemsolving techniques to discover complex patterns ii. describe patterns as general rules consistent with findings iii. verify the validity of these general rules. The student is able to: i. write down the missing exact values in the tables (Q11), use these values to sketch the graphs (Q12), and apply the addition / subtraction selected (Q13) to find the missing values (Q14) ii. apply the selected values to sketch the graphs (Q15), describe what they notice (Q16), and suggest a general rule about the relationship between the graph of 𝑦 = sin(𝑥) / 𝑦 = cos(𝑥) and the graph of 𝑦 = sin(𝑥 − 𝐶) / 𝑦 = cos(𝑥 − 𝐶), for both 𝐶 > 0 and 𝐶 < 0 (Q17). iii. verify their general rule by applying it to the graphs (Q18)(a), by writing down the missing exact values (Q18)(b), and by plotting the points from the table and sketching the graphs (Q18)(c). The student is able to: i. select and apply mathematical problemsolving techniques to discover complex patterns ii. describe patterns as general rules consistent with correct findings iii. prove, or verify and justify, these general rules. The student is able to: i. discover a relationship between the graph of 𝑦 = sin(𝑥) / 𝑦 = cos(𝑥) and the graph of 𝑦 = sin(𝐵𝑥) / 𝑦 = cos(𝐵𝑥), for any real value of 𝐵 7-8 5-6 3-4 1-2 0 Task Specific Descriptor (Q18)(a) ii. iii. describe their relationship as general rule (Q18)(b) verify and justify their general rule (Q18)(c). © 7on7 Academy 2023 Assessment Criterion C: Communicating Achievement Level Descriptor (MYP5) 7-8 5-6 3-4 1-2 0 Task Specific Descriptor The student does not reach a standard described by any of the descriptors below. The student is able to: The student is able to: i. use limited mathematical language i. use mathematical vocabulary poorly ii. use limited forms of mathematical ii. present their findings in an unclear or limited way representation to present information iii. (not demonstrated at this level) iii. (not demonstrated at this level) iv. communicate their findings in a way that is difficult iv. communicate through lines of reasoning that to follow are difficult to interpret v. (not demonstrated at this level). v. (not demonstrated at this level). The student is able to: The student is able to: i. use some appropriate mathematical i. use mathematical vocabulary somewhat clearly language ii. present their findings in a somewhat clear or ii. use appropriate forms of mathematical somewhat limited way representation to present information iii. (not demonstrated at this level) adequately iv. communicate their findings completely iii. (not demonstrated at this level) v. communicate their findings in a way that is iv. communicate through lines of reasoning that somewhat organized. are complete v. adequately organize information using a logical structure. The student is able to: The student is able to: i. usually use appropriate mathematical i. use mathematical vocabulary usually correctly language ii. present their findings mostly clearly and correctly ii. usually use appropriate forms of iii. present their findings using different forms of mathematical representation to present representation information correctly iv. communicate their findings completely and iii. usually move between different forms of mathematical representation coherently, but not concisely iv. communicate through lines of reasoning that v. communicate their findings in a way that is mostly are complete and coherent organized. v. present work that is usually organized using a logical structure. The student is able to: The student is able to: i. consistently use appropriate mathematical i. use mathematical vocabulary consistently language correctly ii. use appropriate forms of mathematical ii. present their findings consistently clearly and representation to consistently present correctly information correctly iii. present their findings effectively using different iii. move effectively between different forms of forms of representation mathematical representation iv. communicate their findings completely, iv. communicate through lines of reasoning that coherently, and concisely are complete, coherent and concise v. communicate his/her findings in a way that is v. present work that is consistently organized organized and follows a logical structure. using a logical structure. © 7on7 Academy 2023 Introduction The trigonometric functions, also called circular functions, are functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, mechanics, geodesy, and many others. The most widely used trigonometric functions are the sine and cosine functions, whose graphs are shown below. Sine function: 𝒚 = 𝐬𝐢𝐧(𝒙) Cosine function: 𝒚 = 𝐜𝐨𝐬(𝒙) In this assessment task we work through scenarios of changing the function rule of the sine and cosine functions by addition and/or multiplication and see how such change affects their graph. Let’s explore! Tasks The graph of 𝑦 = sin(𝑥) and the graph of 𝑦 = cos(𝑥) are shown above and are also lightly sketched on the coordinate planes provided below. (1) Use your GDC (Graphic Display Calculator) to sketch, on the coordinate planes provided, the graph of a. 𝑦 = sin(𝑥) + 1, c. 𝑦 = cos(𝑥) − 1, b. 𝑦 = sin(𝑥) − 2, d. 𝑦 = cos(𝑥) + 2. [B: 1-2, i] © 7on7 Academy 2023 (2) Briefly describe what you notice between the lightly graphed sine/cosine function and the graph you sketched: how does the addition/subtraction affect the graph of 𝑦 = sin(𝑥) and the graph of 𝑦 = cos(𝑥)? [B: 1-2, i] (3) State your prediction for the rule you observed in question (1) by selecting one of the underlined words in each part of the statement below. “The graph of 𝑦 = sin(𝑥) + 𝐷 is the result of a translation / reflection / rotation / dilation of the graph of 𝑦 = sin(𝑥) by 𝐷 units vertically / horizontally.” [B: 1-2, ii] (4) Write down a similar statement to state your prediction about the relationship between the graph of 𝑦 = cos(𝑥) + 𝐷 and the graph of 𝑦 = cos(𝑥). [B: 1-2, ii] The graph of 𝑦 = sin(𝑥) and the graph of 𝑦 = cos(𝑥) are once again lightly sketched on the coordinate planes provided below. (5) Based on your observation(s) described in question (2) above, and without the help of your GDC, sketch the graph of a. 𝑦 = sin(𝑥) + 2, b. 𝑦 = cos(𝑥) − 2. [B: 1-2, ii] © 7on7 Academy 2023 Similarly to the previous coordinate planes, the graph of 𝑦 = sin(𝑥) and the graph of 𝑦 = cos(𝑥) are lightly sketched on the coordinate planes provided below. (6) Use your GDC to sketch, on the coordinate planes provided, the graph of a. 𝑦 = 2 sin(𝑥), b. 𝑦 = 3 cos(𝑥). [B: 3-4, i] (7) Briefly describe what you notice between the lightly graphed sine/cosine function and the graph you sketched: how does the multiplication affect the graph of 𝑦 = sin(𝑥) and the graph of 𝑦 = cos(𝑥)? [B: 3-4, i] (8) Similarly to questions (1) and (5), use your GDC to sketch, on the coordinate planes provided, the graph of a. 𝑦 = − sin(𝑥), b. 𝑦 = − cos(𝑥). [B: 3-4, i] © 7on7 Academy 2023 (9) Briefly describe what you notice between the lightly graphed sine/cosine function and the graph you sketched: how does the multiplying by −1 affect the graph of 𝑦 = sin(𝑥) and the graph of 𝑦 = cos(𝑥)? [B: 3-4, i] (10) Similarly to questions (3) and (4), suggest a general rule about the relationship a. between the graph of 𝑦 = 𝐴 sin(𝑥) and the graph of 𝑦 = sin(𝑥), and b. between the graph of 𝑦 = 𝐴 cos(𝑥) and the graph of 𝑦 = cos(𝑥). Your general rule should make a distinction between 𝐴 > 0 and 𝐴 < 0. [B: 3-4, ii] The table below shows the exact values of two trigonometric functions (sine and cosine) for angles 0°, 30°, 45°, 60°, and 90°, as well as the radian equivalents of these angle measures. Degree 𝟎° 𝟑𝟎° 𝟒𝟓° 𝟔𝟎° 𝟗𝟎° Radian 0 𝜋 6 𝜋 4 𝜋 3 𝜋 2 Sine 0 1 2 √2 2 √3 2 1 Cosine 1 √3 2 √2 2 1 2 0 Table 1 © 7on7 Academy 2023 (11) Use the values of the trigonometric functions in the table above, your knowledge of unit circles, and the graphs of 𝑦 = sin(𝑥) and 𝑦 = cos(𝑥) to write down the missing exact values in the tables below. Some of the values have been included for you. Degree −𝟏𝟖𝟎° −𝟏𝟓𝟎° −𝟏𝟑𝟓° −𝟏𝟐𝟎° −𝟗𝟎° −𝟔𝟎° −𝟒𝟓° −𝟑𝟎° Radian −𝜋 Sine − − 1 2 2𝜋 3 − 𝜋 4 −1 Cosine − − 1 2 1 2 √2 2 Table 2 Degree 𝟏𝟐𝟎° Radian 2𝜋 3 𝟏𝟑𝟓° 𝟏𝟓𝟎° 𝟏𝟖𝟎° 𝟐𝟐𝟓° 𝜋 √2 2 Sine 𝟐𝟏𝟎° Cosine − − 1 2 √3 2 − √2 2 Table 3 Degree 𝟐𝟒𝟎° Radian 4𝜋 3 Sine Cosine 𝟐𝟕𝟎° 𝟑𝟎𝟎° 𝟑𝟏𝟓° 𝟑𝟑𝟎° 𝟑𝟔𝟎° 7𝜋 4 −1 − 1 2 1 2 1 Table 4 [B: 5-6, i] © 7on7 Academy 2023 (12) Without the help of your GDC, use the values in the tables above to sketch the graph of a. 𝑦 = sin(𝑥), b. 𝑦 = cos(𝑥). [B: 5-6, i] (13) Select a value from the list provided below. −60° −30° 60° 30° [B: 5-6, i] (14) Using the exact values in the tables above, apply the addition/subtraction to the value of your choice from question (13) to find the missing values in the tables below. 𝒙 𝑥+ −𝟗𝟎° −𝟔𝟎° −𝟑𝟎° 𝟎° 𝟑𝟎° 𝟔𝟎° 𝟗𝟎° 𝟏𝟐𝟎° 𝟏𝟓𝟎° ° °) sin(𝑥 + Table 5 𝒙 𝑥− cos(𝑥 − −𝟗𝟎° −𝟔𝟎° −𝟑𝟎° 𝟎° 𝟑𝟎° 𝟔𝟎° 𝟗𝟎° 𝟏𝟐𝟎° 𝟏𝟓𝟎° ° °) Table 6 [B: 5-6, i] © 7on7 Academy 2023 (15) Using the coordinate planes, apply your selected value to sketch the graph of a. 𝑦 = sin(𝑥 + ° ) b. 𝑦 = cos(𝑥 − ° ) [B: 5-6, ii] (16) Briefly describe what you notice a. between the graph of 𝑦 = sin(𝑥) in question (12)(a) and the graph of the function you sketched in question (15)(a), and [B: 5-6, ii] b. between the graph of 𝑦 = cos(𝑥) in question (12)(b) and the graph of the function you sketched in question (15)(b), and. [B: 5-6, ii] © 7on7 Academy 2023 (17) Similarly to questions (3), (4), and (10), suggest a general rule about the relationship a. between the graph of 𝑦 = sin(𝑥) and the graph of 𝑦 = sin(𝑥 − 𝐶), and b. between the graph of 𝑦 = cos(𝑥) and the graph of 𝑦 = cos(𝑥 − 𝐶). Your general rule should make a distinction between 𝐶 > 0 and 𝐶 < 0. [B: 5-6, ii] (18) Verify that your general rule works a. by applying your rule suggested above to the graphs of 𝑦 = sin(𝑥) and 𝑦 = cos(𝑥), 𝑦 = sin(𝑥 − 60°) 𝑦 = cos(𝑥 + 120°) b. by writing down the missing exact values in the tables below, and 𝒙 −𝟗𝟎° 𝑥 − 60° −150° sin(𝑥 − 60°) − −𝟔𝟎° −𝟑𝟎° 𝟎° 𝟑𝟎° 𝟔𝟎° 𝟗𝟎° 𝟏𝟐𝟎° −30° 1 2 − 𝟏𝟓𝟎° 90° 1 2 1 Table 7 𝒙 −𝟗𝟎° −𝟔𝟎° −𝟑𝟎° 𝟎° 𝟑𝟎° 𝑥 + 120° 30° 150° cos(𝑥 + 120°) √3 2 − √3 2 𝟔𝟎° 𝟗𝟎° 𝟏𝟐𝟎° 𝟏𝟓𝟎° 270° 0 Table 8 © The EdVaults – All Rights Reserved c. by plotting the points from the table of values and connect them to sketch the graph of 𝑦 = sin(𝑥 − 60°) and the graph of 𝑦 = cos(𝑥 + 120°). 𝑦 = sin(𝑥 − 60°) 𝑦 = cos(𝑥 + 120°) [B: 5-6, iii] (19) In this final part of the assessment task, use a method of your own to a. explore a relationship i. between the graph of 𝑦 = sin(𝑥) and the graph of 𝑦 = sin(𝐵𝑥), and ii. between the graph of 𝑦 = cos(𝑥) and the graph of 𝑦 = cos(𝐵𝑥) for any real value of 𝐵, but specifically mentioning 𝐵 = −1, b. describe your relationship as a general rule, and c. verify and justify your general rule by applying it to examples of your own. In your work, make sure to • use appropriate mathematical language, • describe findings clearly and correctly, and • present work in a neat and organized manner. [B: 7-8, i-iii] Use the space on the next page to carry out your investigation and record your findings. © The EdVaults – All Rights Reserved Show your work here for question (19). You may use the coordinate planes below. © The EdVaults – All Rights Reserved