MEI GeoGebra Tasks for A2 Core

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1 Task 1: Functions The Modulus Function 1. Plot the graph of y = x : use y = x or y = abs(x) 2. Plot the graph of y = ax+b : use y = ax + b or y = abs(ax+b) If prompted click Create Sliders. What combination of transformations maps the graph of y = x onto the graph of y = ax+b? Where is the vertex on the graph of y = ax+b? Where does the graph of y = ax+b intersect the y-axis? Problem (Try the question with pen and paper first then check it on your software) Sketch the graph of y = 3x+2 3 and find the points of intersection with the axes. Investigate the graphs of - y = f(x) - y = f( x ) for different functions f(x), e.g. f(x) = sin(x) or f(x) = x³ x². Investigate the solutions to the inequality x + a +b > 0.

2 Task 2: Inverse functions 1. Enter the function 2 f( x) x 1 : f(x) = x Plot the inverse function: Invert(f) What graphical transformation maps the graph of the original function onto its inverse? Why is the inverse function only defined for part of the original function? Can you confirm algebraically that the function given for g(x) is the inverse function? Try finding the inverses of some other functions. Problem (Check your answers by plotting the graphs on your software) Find inverses of the following functions: f( x) ( x 3) 2 g( x) x 3 h( x) 1 x 2 Find the inverse of the function a quadratic function 2 f( x) x 6x 7 2 f( x) ax bx c?. Can you always find the inverse of Investigate the graphs of the inverse trigonometric functions (you might find radians more convenient for this).

3 Task 3: Trigonometry Double Angle formulae 1. Plot the curve: y = sin(x) cos(x) 2. Plot the curve: y = a sin(bx) If prompted click Create Sliders. Use a space between a and sin. Don t type asin(x) as this is the inverse sine. For what values of a and b does sin xcos x asin bx? Find values of a, b and c so that: o cos² x = a cos (bx) + c o sin² x = a cos (bx) + c How do these relationships link to the double angle formulae for sin and cos? Problem (Try the question with pen and paper first then check it on your software) Solve sin 2 cos 0 in the range tan Describe the relationship tan 2 graphically. 2 1 tan Find expressions for sin3 and cos3 in terms of sin and cos.

4 Task 4: Trigonometry: Rcos(θ α) 1. Plot the curve: y = cos(x) + 2sin(x) 2. Plot the curve: y = R cos(x α) Use a space between R and cos. Can you find values of α and R so that the curves are the same? Can you find values for α and R for any a and b where acos x bsin x Rcos( x )? Can you explain the relationship using Rcos( x ) Rcos xcos Rsin xsin? Problem (Check your answer by plotting the graphs on your software) Express 4cos 3sin in the form R cos( ) where 0. 2 Explore how the form R cos( ) can be used to find the maximum value of acos bsin and the angle at which it occurs. Investigate the height of a rectangle as it is rotated through an angle θ about one of its corners.

5 Task 5: Differentiation Trigonometric functions 1. Plot the curve: y = sin(x) Use d from the f(x) dx keyboard or type f (x) 2. Plot the gradient function by entering Derivative(f) in the input bar. How does the gradient function relate to the original graph of y = sin(x): o What are its maximum and minimum values? o When is the gradient function 0? o For what values of x do these (max, min or 0) occur? Can you suggest a function for the derivative of y = cos(x)? Problem (Check your answer by plotting the graph and the tangent on your software) Find the equation of the tangent to the curve y = sin x at the point x. 3 Investigate the derivatives of y = sin ax and y = b sin x. Explain why this wouldn t work as neatly if the angle was measured in degrees.

6 Task 6: Points of Inflection 1. Plot the curve y = x³ 3x² + x + 3 Use d from the f(x) dx keyboard or type f (x) 2. Plot the gradient function by entering Derivative(f) in the input bar. 3. Use the Point tool to add a point A on the curve 4. Use the Tangent tool to add a tangent to the curve at A 5. Move the point A to (1, 2). What is special about the point (1, 2), i.e. what are its unique features? Can you find the equivalent point for some other cubics? Problem (Check your answer by plotting the graph and the tangent on your software) Find the point of inflection of curve with equation y x x x Find some example of cubic functions that have no stationary points. Will these have points of inflection? Use Inflection(f). Will a quartic function always, sometimes or never have a point of inflection?

7 Task 7: Gradients of tangents to the natural logarithm y=lnx 6. Plot the curve y = ln(x) 7. Use the Point tool to add a point A on the curve y = ln x 8. Use the Tangent tool to add a tangent to y = ln x at A 9. Use the Slope from the Measure tools to find the gradient of the tangent, m 10. Enter the point B=(x(A),m) 11. Add a trace to the point B What is the relationship between the point and the gradient of the tangent on y = ln x? How does this relationship change for the graphs of y = ln 2x, y = ln 3x? Problem (Check your answer by plotting the graph and the tangent on your software) Find the equation of the tangent to the curve y = ln x at the point x 2. Find the tangent to y = ln x that passes through the origin. Explain the relationship between the derivatives of y = e x and y = ln x. Hint: consider the point (a,b) on y = e x and the reflected point (b,a) on y = ln x

8 Task 8: Tangents to parametric curves 1. Plot the parametric curve x = t², y = t³: (t², t³) 2. Plot the point: (p², p³) If prompted click Create Sliders. 3. Use the Tangent tool to add a tangent to the curve at A 4. Use the Slope from the Measure tools to find the gradient of the tangent What is the relationship between d y dx, d x dt and d y dt? Does this relationship for other parametric curves? 1 e.g. x 2t 1, y or x cos t, y sin t. t Problem (Check your answer by plotting the graph and the tangent on your software) Find the coordinates of the points on the curve x 2t cos t, y sin t, 2 t 2 for which the tangent to the curve is parallel to the x-axis. Explore how you can find the equation of the tangent to a parametric curve at a point. Describe how to find the tangent to a parametric curve that passes through a specific point that is not on the curve.

9 Task 9: Partial Fractions 5x 1 1. Enter the function y : y = (5x 1)/((x+1)(x 2)) ( x1)( x2) A B 2. Enter the function y : y = a/(x+1) + b/(x 2) x1 x 2 If prompted click Create Sliders. Find values of a and b so that the graphs of the functions are the same. Question for discussion How could you find the values using 5x 1 A B ( x 1)( x 2) x 1 x 2? Does this method work for 2x 7 A B ( x 2)( x 3) x 2 x 3? Problem (Check your answers by plotting the graphs on your software) Find values of A and B so the following can be expressed as partial fractions: 7x 14 A B ( x 3)( x 4) x 3 x 4 Find A, B and C such that Find A, B and C such that x x A B C ( x 1)( x 3) x 1 x 3 ( x 3) x 3x 7 A Bx C 2 2 ( x 2)( x 3) x 2 x 3

10 Task 10 Numerical Methods: Change of sign This task requires GeoGebra Classic Solving x x x : 1. Plot the function f( x) x x 4x 1 : f(x) = x 3 + x 2 4x +1 In this example you can see that there f( x) 0 has roots that lie between: x = 3 and x = 2 x = 0 and x = 1 x = 1 and x = 2 2. Enable the spreadsheet: View > Spreadsheet 3. In cells A1 and A2 enter the values 1, 1.1 and then fill down to In cell B1 enter f(a1) and then fill down. In this example there is a change of sign between x = 1.3 and x = You can now investigate further by changing the column A so that it goes from 1.3 to 1.4 in steps of You can check your answer in the Graphics view by using the roots tool. NB The number of decimal places can be set with Options > Rounding. Problem Use your software to find the other roots of this equation using the change-of-sign method. Use your software to find the roots of other equations using the change-of-sign method.

11 Task 11 Numerical Methods: Fixed Point Iteration This task requires GeoGebra Classic Solving x x x : 2. Plot the function f( x) x x 4x 1 : f(x) = x 3 + x 2 4x +1 In this example you can see that there f( x) 0 has roots that lie between: x = 3 and x = 2 x = 0 and x = 1 x = 1 and x = 2 3. Enable the spreadsheet: View > Spreadsheet 4. To add the recurrence relation enter the function g( x) x 4x 1 : g(x) = cbrt( x 2 + 4x 1) 5. In Cell A1 enter 0 and in Cell B1 enter 1. x n 1 x n 4x n 1, 6. In Cell A2 enter A1+1 and in cell B2 enter g(b1) and fill down. 7. You can check your answer in the Graphics view by using the roots tool. NB The number of decimal places can be set with Options > Rounding. Problem Use your software to find the other roots of this equation using a fixed-point iteration. You will need to use other rearrangements of the equation f( x) 0 in the form x g( x). Use your software to find the roots of other equations using a fixed-point iteration.

12 Task 12 Numerical Methods: Newton-Raphson Method This task requires GeoGebra Classic Solving x x x : 1. Plot the function f( x) x x 4x 1 : f(x) = x 3 + x 2 4x +1 In this example you can see that there f( x) 0 has roots that lie between: x = 3 and x = 2 x = 0 and x = 1 x = 1 and x = 2 2. Enable the spreadsheet: View > Spreadsheet f( xn ) 3. To add the recurrence relation x n 1 x f '( xn ), enter : g(x) = x f(x)/f (x) 4. In Cell A1 enter 0 and in Cell B1 enter In Cell A2 enter A1+1 and in cell B2 enter g(b1) and fill down. 6. You can check your answer in the Graphics view by using the roots tool. NB The number of decimal places can be set with Options > Rounding. Problem Use your software to find the other roots of this equation using the Newton-Raphson method. Use your software to find the roots of other equations using the Newton-Raphson method.

13 Task 13: Sequences/Series Sum of an Arithmetic Progression Example: 5 2n 1. n1 This task requires GeoGebra Classic 1. Input the function f(x)=2x+1 2. Enable the spreadsheet: View > Spreadsheet 3. In column A enter: 1, 2, 3, 4, 5 4. In Cell B1 enter =f(a1) and fill-down 5. In Cell C1 enter =Sum(B1:B5) Now find the sum of the terms of some other arithmetic progressions. Why will the terms of bn+c (for n = 1,2,3, ) be an arithmetic progression (AP)? How can you express the link between the terms of an AP and its sum? Problem (Try the question with pen and paper first then check it on your software) What is the first term of an arithmetic progression if the 3 rd term is 11 and the sum of the first 10 terms is 185? Investigate arithmetic progressions with the same sum, e.g. how many APs can you find that have a sum of 100? Express the n th triangle number as the sum of an AP. Investigate whether the n th hexagonal number can be expressed as the sum of an AP.

14 Task 14: Sequences/Series Sum of an Geometric Progression This task requires GeoGebra Classic Example: 5 n1 3 2 n 1. Input the function f(x)=3*2^x 2. Enable the spreadsheet: View > Spreadsheet 3. In column A enter: 1, 2, 3, 4, 5 4. In Cell B1 enter =f(a1) and fill-down 5. In Cell C1 enter =Sum(B1:B5) Now find the sum of the terms of some other geometric progressions. Why will the terms of b c n (for n = 1,2,3, ) be a geometric progression (GP)? How can you express the link between the terms of a GP and its sum? Problem (Try the question with pen and paper first then check it on your software) What is the common ratio of a geometric progression if the 2 nd term is 30 and the sum of the first 6 terms is 3640? Investigate the sums of geometric series where the common ratio is negative and/or less than 1. Investigate how savings or loans can be expressed as the sum of a geometric progression.

15 Task 15: Trigonometry Solving equations in radians 1. Plot the function: f(x) = sin(x) 2. Enter the function: g(x) = k If prompted click Create Sliders. 3. In the input bar enter: Intersect(f, g, -2pi, 4pi) What symmetries are there in the positions of the points of intersection? How can you use these symmetries to find the other solutions based on the value of sin -1 x given by your calculator? (This is known as the principal value.) Problem (Try the question just using the sin -1 function on your calculator first then check it using the software) Solve the equation sin x = 0.5 ( 2π < x 4π), giving your answers in terms of π. Investigate the symmetries of the solutions to cos x = k and tan x = k. Investigate the symmetries of the solutions to sin 2x = k.

16 Using these tasks MEI GeoGebra Tasks for A2 Core Teacher guidance These tasks are designed to help students in understanding mathematical relationships better through exploring dynamic constructions. They can be accessed using the computerbased version of GeoGebra or the tablet/smartphone app. Each task instruction sheet is reproducible on a single piece of paper and they are designed to be an activity for a single lesson or a single homework task (approximately). Most of these tasks have been designed with the following structure Construction: step-by-step guidance of how to construct the objects in GeoGebra. Students will benefit from learning the rigorous steps need to construct objects and this also removes the need to make prepared files available to them. If students become confident with using GeoGebra they can be encouraged to add additional objects to the construction to aid their exploration. : This discussion can either be led as a whole class activity or take place in pairs/small groups. The emphasis is on students being able to observe mathematical relationships by changing objects on their screen. They should try to describe what happens, and explain why. Problem: Students are expected to try the problem with pen and paper first then check it on their software. The purpose is for them to formalise what they have learnt through exploration and discussion and apply this to a standard style question. Students could write-up their answers to the discussion questions and their solution to this problem in their notes to help consolidate their learning and provide evidence of what they ve achieved. This problem can be supplemented with additional textbook questions at this stage if appropriate. : Extension activities with less structure for students who have successfully completed the first three sections. Task 1: The Modulus Function Students should consider how this relates to the graph of y = ax+b 5 1 Problem solution: x, y Students might need some help structuring the investigation into x + a +b > 0. One strategy is to fix either a or b and investigate changing the other parameter. Task 2: Inverse functions The aim of this task is to reinforce the link between the reflection in the line y = x and rearranging y f( x) to express x in terms of y. Problem solutions:

17 1 1 3 f ( x) x 3 g ( x) x h ( x) 2 x It is important to emphasise that the domain of the original function needs to be restricted so that it is one-to-one for the inverse to be a function. 1 1 The invert[f] command will only work on functions with a single instance of x so students should compare the completed square form to the expanded form of a quadratic. Task 3: Trigonometry Double Angle formulae Students might need some help structuring the investigation into sin x cos x = a sin (bx). One strategy is to fix b and investigate changing a first to find a curve with the correct amplitude. Use of the compound angle formulae for sin( a b) and cos( a b) might be useful for some students to verify their results. 5 3 Problem solution:,,, Task 4: Trigonometry: Rcos(θ α) Students are expected to be able to relate their findings to the expansion of Rcos( x A) Rcos xcos A Rsin xsin A. Problem solution: 4cos 3sin 5cos( 0.644). Task 5: Differentiation Trigonometric functions By considering key points the students should be able to observe that this has the same shape as cos(x). Problem solution: x 3 y or y0.5x Task 6: Points of Inflection The point (1, 2) is a (non-stationary) point of inflection on y x 3x x 2. Students should be encourage to look at the geometry of the curve, its tangent and its derivative and observe some of the following: it is where the curve changes from concave to convex; it is the point where the tangent slopes down the most; it is at the same x-value as when the derivative has a local minimum; it is the only point where the tangent crosses the curve (at that point); it is a point of rotational symmetry (this would not necessarily be the case for curves other than cubics).

18 Problem solution: The point of inflection of y x x x is at ( 2, 4). Task 7: Gradients of tangents to the exponential function y=ln x This task can be done on its own or with task 6. The aim of this task is for students to be able to find the gradients and equations of tangents to the natural logarithm function. For the second discussion point students might be surprised that the result doesn t change but they should be encouraged to think of this in terms of laws of logs. Problem solution: y0.5x Task 8: Tangents to parametric curves Students might find it useful to make a table of values for d y dx, d x and d y. Students should dt dt be encouraged to explore further parametric equations and to consider their relationship in dy dy terms of dt dx dx. dt Problem solution: 3 t, : points are (2,1),(2, 1). 2 2 Task 9: Partial Fractions This task can be used as an introduction to partial fractions or as a consolidation exercise. Students should be encouraged to express their methods algebraically. Solutions to partial fractions: 5x ( x 1)( x 2) x 1 x 2 7x ( x 3)( x 4) x 3 x 4 2x ( x 2)( x 3) x 2 x 3 2 7x 29x ( x 1)( x 3) x 1 x 3 ( x 3) x 3x 7 x ( x 2)( x 3) x 2 x 3

19 Task 10: Numerical Methods Change of sign This task is a set of instructions for how to implement the change of sign method on the software. Students are encouraged to work through these instructions and then try finding the other roots of this equation. The roots are x 2.651, x 0.274, x It is useful to have some additional equations for students to be finding the roots of once they have completed this sheet. Task 11: Numerical Methods Fixed Point Iteration This task is a set of instructions for how to implement the change of sign method on the software. Students are encouraged to work through these instructions and then try finding the other roots of this equation. The roots are x 2.651, x 0.274, x The rearrangement x x 4x 1 will find the first and third of these roots. An alternative x x 1 rearrangement, such as x is required to find the middle root. 4 It is useful to have some additional equations for students to be finding the roots of once they have completed this sheet. Task 12 Numerical Methods: Newton-Raphson Method This task is a set of instructions for how to implement the Newton Raphson method on the software. Students are encouraged to work through these instructions and then try finding the other roots of this equation. The roots are x 2.651, x 0.274, x It is useful to have some additional equations for students to be finding the roots of once they have completed this sheet. Task 13: Sum of an Arithmetic Progression This task reinforces the terms of a sequence and what the sigma notation is showing. Before moving on to the problem you should show the two forms of the sum of an AP. Problem solution: 5 Task 14: Sum of a Geometric Progression This task reinforces the terms of a sequence and what the sigma notation is showing. It is unlikely that many students will be able to derive a general formula for the sum of a GP without assistance. You will need to show them the sum of a GP before they attempt the problem and you can link their observations to the formula and discuss whether they are consistent with it. Problem solution: 3

20 Task 15: Trigonometry Solving equations in radians This task encourages students to think about the symmetries of the trigonometric graphs and use these in finding solutions to equations. Problem solution: x,,,,,