advertisement

MEI Conference 2014 Using GeoGebra in A level Core Tom Button [email protected] MEI GeoGebra Tasks for AS Core Coordinate Geometry: Perpendicular lines 1. Use New Point (2nd menu) to add a point, A. 2. Use New Point (2nd menu) to add a point, B. 3. Use Line (3rd menu) to create the line through A and B. 4. Use New Point (2nd menu) 5. Use Perpendicular Line (4th menu) perpendicular to the line AB. Click on the point C and then the line. to add a point, C. to create the line through C and You can display the gridlines by clicking the gridlines icon in Graphics style bar. Questions What is the relationship between the equations of the lines? What is the relationship between the equations of the lines then they are written in the form y = mx + c? Problem Show that the line perpendicular to the line through (5,1) and (1,3) that passes through the point (3,4) has equation y = 2x – 2. Further Tasks For two points A and B what are the possible positions for C so that the line through C is a perpendicular bisector? For three points A, B and C find the point of intersection of the two lines. www.mei.org.uk/geogebra MEI GeoGebra Tasks for AS Core Algebra: Quadratic equations 1. Use New Point (2nd menu) 2. In the input bar enter: a=x(A) b=x(B) to add two new points on the x-axis, A and B. Enter these separately and press enter after each one. 3. In the input bar enter: y=(x-a)(x-b) 4. Use New Point (2nd menu) to add a new point (not on either axis), C. 5. In the input bar enter: p=x(C) q=y(C) Enter these separately and press enter after each one. 6. In the input bar enter: y=(x-p)^2+q You can display the gridlines by clicking the gridlines icon in Graphics style bar. Questions Can you find positions for A, B and C so that the two graphs are the same? What is the relationship between the values of a, b, p and q when the graphs are the same? Problem Solve the equation x² – 2x – 8 = 0 by both factorising and completing the square. Further Tasks Add a Slider (11th menu) and set its name to k. Change the equation in step 7 to y=k(x-p)^2+q Where does this curve cross the x-axis? Can you change the equation in step 4 so the curves are the same? www.mei.org.uk/geogebra MEI GeoGebra Tasks for AS Core Differentiation: Exploring the gradient on a curve 1. In the input bar enter a cubic function: e.g. f(x)=x^3-2x^2-2x+1 2. Use New Point (2nd menu) to add a point on the curve. 3. Use Tangent (4th menu) to create a tangent to the curve at point A. 4. Use Slope (8th menu) to measure the slope of the tangent. 5. Plot the gradient function by entering g(x)=f '(x) in the input bar. You might find it easier to see if you change the gradient function to a red dotted line using the Graphics Styling bar. Question How is the gradient of the slope (as the point moves) related to shape of the gradient graph? Verify your comments by trying some other functions for f(x). Problem Can you find functions that have the following gradient functions: Further Tasks Describe the gradient graph for cubics that have 0, 1 and 2 stationary points. Investigate the minimum (or maximum) point on the gradient graph for a cubic. www.mei.org.uk/geogebra MEI GeoGebra Tasks for AS Core Integration: Area under a curve 1. In the Input bar enter: f(x)=x^2 2. Use Slider (11th menu) It is essential that this is entered as a function f(x). to create a slider for a. 3. In the Input bar enter: A=Integral[f, 0, a] Create the slider with minimum value 0. Questions What is the relationship between the area and the value of a? What is the relationship if f(x) is changed to a different power of x? Problem Find the area under f(x) = x5 between x = 0 and x = 3. Further Tasks Add a Slider (11th menu) for b. Investigate the area under f(x)= xn between x = a and x = b. Investigate the areas under functions that are the sums of powers of x: e.g. f(x)=x³ + 3x² + 4x +1 www.mei.org.uk/geogebra MEI GeoGebra Tasks for AS Core Functions: Transformations 1. Use Slider (11th menu) to create sliders for a and b. 2. In the Input bar enter: f(x)=x^2 It is essential that this is entered as a function f(x). 3. In the Input bar enter: g(x)=f(x+a)+b Questions What transformation maps f(x) onto g(x)? Does this work if other functions are entered for f(x)? Problem Show that f(x) = x4 – 8x³ + 24x² – 32x +13 can be written in the form (x+a)4 + b and hence find the coordinates of the minimum point on the graph of y = f(x). Further Tasks Use Slider (11th menu) to create sliders for c and d. In the Input bar enter: h(x)=c*f(x*d). What transformation maps f(x) onto h(x)? Investigate g(x) and h(x) for f(x)=log10x. NB this is entered as: f(x)=log10(x) Changing f(x) to f(x)=x³–x might help make it clearer. www.mei.org.uk/geogebra MEI GeoGebra Tasks for AS Core Constructing objects in GeoGebra Testing students’ understanding of ideas and reinforcing generalisation Example 0. Create a two points A and B on the x axis. Construct a quadratic graph that passes through A and B. Use the New Point button to add points A and B fixed to the x-axis. In the Input bar define two variables: a = x(A) and b = x(B). Define a new curve on the Input bar: y = (x–a)(x–b) Ideas for AS Core Mathematics 1. Create two points A and B. Construct a third point C which lies on the line perpendicular to AB passing through A and is twice as far away from A as B is. 2. Create points A, B and C fixed to the x-axis and D fixed to the y-axis. Construct a cubic that passes through A, B, C and D. 3. Create a triangle with one point on the origin and one point on the x-axis. Construct circles centred on each vertex such that all three circles touch each other. 4. Create a graph of a quadratic equation that can be moved by dragging the vertex. a. Construct the tangent to the curve with gradient 2 (that works for the vertex in any position). b. Construct the tangent to the curve with gradient b (that works for the vertex in any position). 5. Draw the graph of a straight line through the origin (NB this must be defined as a function, e.g. f(x)= x or f(x)=2x). Add a point A on the positive x-axis. a. Construct a point B such that the integral of f(x) between A and B is 8. b The GeoGebra function for f ( x)dx is: Integral[f, a, b] a b. Construct a point B such that the integral of f(x) between A and B is d. c. Construct a point B such that the integral of f(x)=mx between A and B is d for any value of m or d. 6. Construct a triangle with sides a and b and angle A that demonstrates the ambiguous case of the Sine rule. 7. Draw the graph of y = ax and add the point A on the curve. Construct a point B based on A that you can use with Trace function to obtain the shape of y = logax. 8. (A challenge!) Create two points A and B. Construct a cubic that has stationary points at A and B. (Hint – the midpoint of A and B may help). www.mei.org.uk/geogebra