String Graphs Part 1 + 2

Transcription

1 String Graphs Part + Answers TI-Nspire Investigation Student 45 min Aim Connect the outcomes of Advanced Strings Graphs Part and Advanced Strings Part using transformation matrices Visualising the Connection Open the TI-Nspire file String Graphs 3. Page. contains a visual of the String Graphs produced in Activit. Two matrices control the location of these string patterns: Rotation matrix Dilation matrix The angle (theta) is associated with the rotational matrix and can be changed using the slider. The dilation matrix dilates in both the x and direction and can be adjusted using the k slider. Adjust the sliders to map the lines and points from activit to the lines and points from activit. Question:. Question:. What is the angle (measured in degrees) required to orient the points and lines from activit back to those from activit? Justif our answer. The line = x makes a 45 o angle with the x axis. The points along the = x line must be mapped back to the x axis, 45 o. Similarl the line = -x makes an angle of 45 o with the axis, the points along this line are mapped back to the axis. What is the dilation factor required to map the points and lines from activit back to those from activit? Justif our answer. The approximate slider value that produces a reasonable mapping is:.7. The point (, ) is units from the origin, this point is mapped to (, ) which is onl units from the origin. The points along the line = x are equall spaced, so too those along the x-axis. A dilation of must therefore be applied to all points. Similarl with the point (-, ) mapping to (, ) and all the points along the line = -x. Texas Instruments 7. You ma cop, communicate and modif this material for non-commercial educational purposes provided all

2 String Graphs Part + - Answers Question: 3. Explain how the rotation and dilation connect the String Graphs activities and. The lines = x and = -x are perpendicular, so too the x and axis. The line = x makes an angle of 45 o with the x axis, so a rotation of 45 o (clockwise) will map the points along the line = x onto the x axis. The point (, ) on the line = x is units from the origin; therefore a dilation must be included with the rotation to map the point (, ) to the point (, ). k The dilation matrix represents a dilation factor k from the axis. The dilation matrix k represents a dilation factor k from the x axis. cos( ) sin( ) The rotational matrix sin( ) cos( ) produces a rotation of in a counter-clockwise direction about the origin. Teacher Notes: To make use of the lists in the spreadsheet the coordinate pairs are set in rows rather than columns therefore requiring the rotation matrix to be transposed. Students ma look at the formula structure and wonder wh it appears differentl than on the question sheet. Navigate to problem, page. In this Notes application the angle, dilation and coordinates can be edited (press Enter after each edit). The corresponding matrix entries will automaticall update. The image opposite shows one of the original points in Activit (-, ) transformed via a rotation in a clockwise direction (-45 o ) and dilated b a factor of from both the and x axis. The resulting coordinate (, ) corresponds to the first point in Activit. In Activit points on the axis (, ), (, 9) were connected to points along the x axis (, ), (, ). In Activit points along the line x, (-, ), (-9, 9) were connected to points along the line x (, ), (, ) Texas Instruments 7. You ma cop, communicate and modif this material for non-commercial educational purposes provided all

3 String Graphs Part + - Answers 3 Question: 4. Use the matrix transformations on Page. to show that the points in Activit can be transformed to the original points in Activit. Sample entries shown here Question: 5. The same matrix transformations on Page. can be applied to the points of intersection between consecutive lines. The first four points of intersection in Activit are shown below. Determine their corresponding points in Activit using the matrix transformations. Point : 9 8, Point : 78 6, Point 3: 68 4, Point 4: 6, Texas Instruments 7. You ma cop, communicate and modif this material for non-commercial educational purposes provided all

4 String Graphs Part + - Answers 4 Equations The same transformations applied to the points from Activit and can be applied to the lines. cos( ) sin( ) k x x' sin( ) cos( ) k ' Expressions for x and can be determined on the calculator using inverse matrix operations: x k cos( ) sin( ) x' k sin( ) cos( ) ' Navigate to page 3.. The rotation and dilation matrices have alread been entered so the can be copied and pasted as required. Store -45 o in angle Store : = -45 o in the value for k. The matrix transformations can be entered naturall as the are expressed above. Ctrl + C = Cop Ctrl + V = Paste An alternative method is to highlight the required expression and press [Enter] and the expression will be pasted into the active cursor position. Question: 6. Question: 7. Write expressions for x and in terms of x ' and x cos( 45 ) sin( 45 ) ' x sin( 45 ) cos( 45 ) ' x x ' ' ' x ' The linear equation determined in Advanced String Graphs Part, passing through (-, ) and 9 (, ) is given b: x. Use our result from Question 6 to determine the linear equation passing through the points (, ) and (, ) corresponding to the first equation in Advanced String Graphs Part. '. Texas Instruments 7. You ma cop, communicate and modif this material for non-commercial educational purposes provided all

5 String Graphs Part + - Answers 5 x x ' ' and ' x ' 9 Given: x 9 ' x ' ( x ' ') 9 9 ' ' x ' x ' ' x' ' x' Transformed Equation: x Use result from matrix equation. Substitution of result into equation from Activit. Solve equation for. Question: 8. The linear equation determined in Advanced String Graphs Part, passing through (-9, 9) and 7 36 (, ) is given b: x. Use our result from Question 6 to determine the linear equation passing through the points (, 9) and (, ) corresponding to the first equation in Advanced String Graphs Part. x x ' ' and ' x ' 7 36 Given: x 7 36 ' x ' ( x ' ') ' ' x ' x ' 4 ' 8 x' 36 9 ' x' 9 Transformed Equation: 9 x 9 Use result from matrix equation. Substitution of result into equation from Activit. Solve equation for. Question: 9. Navigate to page 4. and enter the appropriate transformations and equation, using the function notation provided: f( x ). a) Check our answers to Questions 7 and 8. Complete on calculator b) Check the following two equations from Part. Complete on calculator Texas Instruments 7. You ma cop, communicate and modif this material for non-commercial educational purposes provided all

6 String Graphs Part + - Answers 6 Question:. x 6 The parabola passing through the points of intersection in Activit was:. Use an appropriate matrix transformation to write an equation for the equation to the curve from Activit. Do not attempt to express the equation in the form =. The equation can however be copied and pasted into the graph application on page. (Relation ) to confirm it is correct. ( x ) 6 x cos( 45 ) sin( 45 ) ' x sin( 45 ) cos( 45 ) ' x x ' ' ' x ' x ( x ) x x x x x Teacher Notes: The equation above can be graphed directl using the relational graphing option. Extension Conic Sections A parabola is defined as a set of points equidistant from a single point (focus) and a line (directrix), that is: d d in the diagram opposite. Question:. x 6 Use the equation from Activit : to determine the location of the focus and directrix. Solve ( ), x f d x f d ( f d) ( f d) x 6 Since:, students can set up two equations and solve simultaneousl. Equations: d f f d 4 d and f and 6 Texas Instruments 7. You ma cop, communicate and modif this material for non-commercial educational purposes provided all

7 String Graphs Part + - Answers 7 Teacher Notes: The Graph Menu: Analse Graphs > Analse Conics > Foci will provide the same result as above, similarl Analse Graphs > Analse Conics > Directrix will produce the other required result. Teachers ma choose to include use the distance propert to determine for this question. Question:. Check our answer to the previous question using a selection of points on the curve. Answers will var: 6 The simplest point to check first is the turning point:, which is the focus and directrix. units from both Students can also do an approximate check using the calculator b placing a point on the curve and measuring the distance from the focus to the point on the curve and also the point on the curve to the directrix. Teacher Notes: Numerous paper folding and geometric constructions can also help students gain a better understanding of the construction of a parabola. A simple activit is to line students up along the front of the classroom. Another student (F) stands approximatel metres from the front of the room and equidistant from either side of the room. The students standing at the front of the room are then instructed to walk forward (perpendicular to the front of the room) and STOP when the believe the are the same distance from the front of the room as the are from student F. The front of the room represents the directrix and student F represents the focus. The curve that students are standing on is a parabola. Texas Instruments 7. You ma cop, communicate and modif this material for non-commercial educational purposes provided all

8 String Graphs Part + - Answers 8 Question: 3. Use transformation matrices to determine the coordinates of the focal point and equation to the directrix for the curve from Activit. Use a selection of points to show that our answer is correct. Original Directrix Equation: x cos( 45 ) sin( 45 ) ' x sin( 45 ) cos( 45 ) ' x x ' ' ' x ' x x New Directrix Equation: x AND 4 Original Focal Point:, x cos( 45 ) sin( 45 ) 4 sin( 45 ) cos( 45 ) 4 x Once again students can use the Graph Menu to analse conics and check their answers. Placing points on the curve and measuring the distance from the focus to the point on the curve and then also from this point to the directrix provides a powerful visual for students to observe and verif that the equation the have graphed is still parabolic. Texas Instruments 7. You ma cop, communicate and modif this material for non-commercial educational purposes provided all