TASK Transforming Functions Question 1 (criteria B) In this question 𝑓(𝑥) = 𝑥 2 + 6𝑥 + 2 a) Show that the turning point of 𝑓(𝑥) is (-3, -7) by completing the square. b) Copy and complete the table below. Function notation Expanded (and simplified) function f(𝑥) =𝑥 2 + 6𝑥 + 2 Turning point (-3, -7) f(𝑥) + 1 f(𝑥) - 6 f(𝑥) + 5 c) Describe a pattern between the function notation f(𝑥) + a and how the turning point changes from the original f(𝑥). d) Predict what the turning point would be for f(𝑥) + 13 Question 2 (criteria B) In this question 𝑓(𝑥) = 𝑥 2 − 5𝑥 − 6 a) Sketch the graph showing any key points (intercepts with axis and the turning point). b) Select your own value(s) of a and sketch the graph 𝑓(𝑥 + 𝑎). c) Identify the relationship between the value of a and the x-intercepts of the graph for 𝑓(𝑥 + 𝑎) compared to the x-intercepts for 𝑓(𝑥). d) Write down a general rule for how the graph 𝑓(𝑥 + 𝑎) is transformed (moved) from 𝑓(𝑥). Question 3 (assessing criteria B and C) In this question, you need to investigate the impact of changing 𝑓(𝑥) has on the graph of quadratic functions. Given 𝑓(𝑥) is a quadratic in the form 𝑓(𝑥) = 𝑎 𝑥 2 + 𝑏𝑥 + 𝑐, investigate what happens to the shape of the parabola for: ● 𝑓(𝑛𝑥) ● -𝑓(𝑥) In each answer you must ● Show clearly how you chose to investigate the shape of the parabola. ● Write down your general rule for each investigation. ● Verify and justify, or prove, each rule. TASK SPECIFIC CLARIFICATION Criterion B: Investigation 7-8 5-6 Task specific clarification The student is able to: i. select and apply appropriate mathematical problem solving techniques to discover complex patterns ii. describe these patterns as general rules consistent with correct findings iii. prove, or verify and justify, these general rules i) Correct approach selected to investigate both of the transformations in Q3. This method is carried out correctly. The student is able to: i. select and apply appropriate mathematical problem solving techniques to discover complex patterns ii. describe these patterns as general rules consistent with findings iii. verify the validity of these general rules i) Correct approach selected to investigate the transformations in Q3. This method is carried out correctly. ii) Correct findings from investigations are summarised as a correct general rule. iii) Rule is checked with an additional example and an explanation for why the graph changes in a certain way is provided OR an algebraic proof clearly shows how f(nx) or -f(x) transforms a graph. ii) Findings from investigations are summarised as a general rule. iii) Rule is checked with an additional example. 3-4 1-2 0 The student is able to: i. apply mathematical problem solving techniques to discover patterns ii. suggest general rules consistent with findings i) f(x) and at least one f(x + a) drawn with a change in x-intercepts noted for question 2. The student is able to: i. apply, with teacher support, mathematical problem solving techniques to discover patterns ii. state predictions consistent with patterns i) Identify a pattern between a and the change in turning point in part 1c) The student does not reach a standard described by any of the descriptors above. ii) A general rule for how ‘a’ affects the graph f(x) is given for question 2. ii) Correctly predict what the turning point will be in part 1d) Criterion C: Communication 7-8 5-6 i. consistently use appropriate mathematical language ii. use appropriate forms of mathematical representation to consistently present information correctly iii. move effectively between different forms of mathematical representation iv. communicate through lines of reasoning that are complete, coherent and concise v. present work that is consistently organized using a logical structure. i. usually use appropriate mathematical language ii. usually use appropriate forms of mathematical representation to present information correctly iii. usually move between different forms of mathematical representation iv. communicate through lines of reasoning that are complete and coherent v. present work that is usually organized using a logical structure. 3-4 i. use some appropriate mathematical language ii. use appropriate forms of mathematical representation to present information adequately iii. communicate through lines of reasoning that are complete iv. adequately organize information using a logical structure. 1-2 i. use limited mathematical language ii. use limited forms of mathematical representation to present information iii. communicate through lines of reasoning that are difficult to interpret. 0 The student does not reach a standard described by any of the descriptors above. Task specific clarification (in question 3 only) i) f(𝑥) notation used correctly Key points of the graph are described using appropriate mathematical language. ii) and iii) words, calculations, graphs, tables, diagrams are used correctly and appropriately - the most effective form of communication is selected at each point. iv) The journey to any rules are clearly shown and explained. No conclusions are written in isolation. Steps and working out are included. v) Question 3 is easy to follow due to the layout. There is a clear plan, investigation and conclusion. ANSWER KEY Question 1 (criteria B) In this question 𝑓(𝑥) = 𝑥 2 + 6𝑥 + 2 a) Show that the turning point of 𝑓(𝑥) is (-3, -7) by completing the square. b) Copy and complete the table below. Function notation Expanded (and simplified) function f(𝑥) =𝑥 f(𝑥) + 1 𝒙 𝟐 + 𝟔𝒙 + 𝟑 (-3, -6) f(𝑥) - 6 𝒙 𝟐 + 𝟔𝒙 − 𝟒 (-3, -13) f(𝑥) + 5 𝒙 𝟐 + 𝟔𝒙 + 𝟕 (-3, -2) 2 + 6𝑥 + 2 Turning point (-3, -7) c) Describe a pattern between the function notation f(𝑥) + a and how the turning point changes from the original f(𝑥). The x-coordinate stays the same, the y-coordinate is -7 + a d) Predict what the turning point would be for f(𝑥) + 13 (-3, 6) Question 2 (criteria B) In this question 𝑓(𝑥) = 𝑥 2 − 5𝑥 − 6 a) Sketch the graph showing any key points (intercepts with axis and the turning point). b) Select your own value(s) of a and sketch the graph 𝑓(𝑥 + 𝑎). c) Identify the relationship between the value of a and the x-intercepts of the graph for 𝑓(𝑥 + 𝑎) compared to the x-intercepts for 𝑓(𝑥). x-intercepts are (-1 - a) and (6 - a) d) Write down a general rule for how the graph 𝑓(𝑥 + 𝑎) is transformed from 𝑓(𝑥). If a is positive, the graph will move left. If a is negative, the graph will move right. 𝒇(𝒙 + 𝒂) is transformed from 𝒇(𝒙) ‘a’ places to the left. Question 3 (assessing criteria B and C) In this question, you need to investigate the impact of changing 𝑓(𝑥) has on the graph of quadratic functions. Given 𝑓(𝑥) is a quadratic in the form 𝑓(𝑥) = 𝑎 𝑥 2 + 𝑏𝑥 + 𝑐, investigate what happens to the shape of the parabola for: ● 𝑓(𝑛𝑥) ● -𝑓(𝑥) In each answer you must ● Show clearly how you chose to investigate the shape of the parabola. ● Write down your general rule for each investigation. ● Verify and justify, or prove, each rule. ● For 𝑓(𝑛𝑥) ● Selected a quadratic function to be 𝑓(𝑥). ● Either sketched the graph or had ‘key values’ in a table. ● Investigated at least one 𝑓(𝑛𝑥) with an ‘n’ of their choice - sketched or table of values. ● Prediction, pattern or rule linking to x-coordinates going from (x, y) to (x/n, y) in xintercepts and turning point. ● Other patterns - y-intercept stays the same. ● General rule - the graph contracts horizontally by scale factor n (shrinks if n>1, stretched if 0<n<1). ● Verify with another value of n ● Example of justification: factorised form which was (x - a)(x - b) becomes (nx-a)(nx-b) so x-intercepts go from a, b to a/n and b/n ● Example of proof. Substituting into the quadratic formula (a becomes an2, b becomes bn) leaves (1/n)quadratic formula. ● For −𝑓(𝑥) ● Selected a quadratic function to be 𝑓(𝑥). ● Either sketched the graph or had ‘key values’ in a table. ● Investigated -𝑓(𝑥) - sketched or table of values. ● Prediction, pattern or rule linking to x-coordinates going from (x, y) to (x,-y) in xintercepts and turning point. ● Other patterns - x-intercepts stay the same ● General rule - the graph is reflected in the x-axis ● Verify with another f(x) and -f(x) ● Example of justification: -f(x) changes the sign of quadratic so would change U shape to n shape ● Example of proof: -(ax2+bx+c) - every x substituted would produce the same y as before, then the negative would give the coordinate (x,-y)