Parabolic Patterns

773 /www/nrich/html/content/01/04/six6/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> <mdo:image src="graph3.gif" alt="Fifteen different wavy functions!"></mdo:image> The illustration shows the graphs of fifteen functions. Two of them have equations $y = x^2$ $y = - (x - 4)^2$ Use a graphic calculator or a graph drawing computer program to sketch these two graphs and then locate them in this illustration. Use the clues given in this information to help you to find the equations of all the other graphs and to draw the pattern of the 15 graphs for yourself. For your solution send in the equations you have found with an explanation of how you did it. What about the equations of these parabolas? <mdo:image height="250" width="592" src="parabolas2.jpg" alt="Graphs y=kx^2"></mdo:image> You may like to use your creative talents to devise your own pattern of graphs and send them to us so that we can base another challenge like this one on the website using your pattern. <div class="framework"> NOTES AND BACKGROUND This sort of challenge is sometimes called an inverse problem because the question is posed the opposite way round to what might have been expected. This is almost like saying: 'here is the answer, what was the question?' Instead of giving the equations of some functions and asking you to sketch the graphs, this challenge gives the graphs and asks you to find their equations. You are being asked to sketch a family of graphs. What makes this a family? All the graphs are obtained by transformations such as reflections and translations of other graphs in the family. The key is to find the simplest function and then tofind transformations of the graph of that function which give the other graphs in the family. If you have access to a graphic calculator, or tograph drawing software, it will not give you the answers. You will have to think for yourself what the equations should be and then the software will enable you to test your own theories and see if you were right.</div> </mdoxml> <mdoxml version="1.0"> We have had solutions from Fiona, a Year 10 student from Stamford High School (Lincolnshire), and from Bei Guo, Kristin and Ryan from Riccarton High School in Christchurch (New Zealand). Well done to you all They all noticed that the parabolas could be treated as three separate sets of five parabolas. Starting with $y = x^2$, the parabola going through the origin, they noticed that it could be raised or lowered to produce the rest of the parabolas in the middle set by either adding or subtracting a $2$ or a $4$. Therefore the solutions for the middle set are: $y = x^2$ $y = x^2 + 2$ $y = x^2 + 4$ $y = x^2 - 2$ $y = x^2 - 4$ Then they used the other equation that had been given, $y = - (x - 4)^2$ , and found that it gave one of the parabolas in the right hand set. $y = - (x^2 )$ is a reflection of $y = x^2$ in the horizontal axis, so that is why the new parabola was an inverted version of the original one. $y = (x - 4)^2$ is a translation of $y = x^2$ by $4$ units to the right, so that is why the new parabola was the inverted parabola shifted $4$ units to the right. As before, they noticed that $y = - (x - 4)^2$ could be raised or lowered to produce the rest of the parabolas in the right hand set by either adding or subtracting a $2$ or a $4$. Therefore the solutions for the right hand set are: $y = - (x - 4)^2$ $y = - (x - 4)^2 + 2$ $y = - (x - 4)^2 + 4$ $y = - (x - 4)^2 - 2$ $y = - (x - 4) ^2 - 4$ Finally, they noticed that the left hand set of parabolas were a reflection of the right hand set in the vertical axis. Therefore, they reasoned that the parabola that went through $(- 4, 0)$ would be $y = - (x + 4)^2$ : the negative sign in front of the brackets produces the inverted parabola, and the $+ 4$ inside the bracket translates it $4$ units to the left. As before, they noticed that $y = - (x + 4)^2$ could be raised or lowered to produce the rest of the parabolas in the left hand set by either adding or subtracting a $2$ or a $4$. Therefore the solutions for the left hand set are: $y = - (x + 4)^2$ $y = - (x + 4)^2 + 2$ $y = - (x + 4)^2 + 4$ $y = - (x + 4)^2 - 2$ $y = - (x + 4)^2 - 4$ </mdoxml> <mdoxml version="1.0"> <h3>Why do this problem ?</h3> In <a href="http://nrich.maths.org/public/viewer.php?obj_id=773&part=">this problem</a>, instead of giving the equations of some functions and asking learners to sketch the graphs, this challenge gives the graphs and asks them to find their equations. This encourages learners to experiment by changing the equations systematically to discover the effect on the graphs. <h3>Possible approach</h3> <div>Start by showing the picture and asking learners to identify the graphs of $y=x^2$ and $y=-(x-4)^2$.</div> <div>Encourage discussion about the similarities and differences between these two graphs, and their equations.</div> <div>Give learners time to experiment with graphical calculators or graphine software with the aim of idenitfying rules for modifying equations to transform graphs and create those on the image.</div> <div>Draw ideas together with the purpose of defining a set of rules that can be applied in other similar situations.</div> <div>A nice extension is for pairs to create a picture of their own, which they then challenge another pair to reproduce.</div> <div>Those learners who are familiar with the idea of completing the square may wish to write the equation of each parabola in this form and consider the coordinates of the vertex. You can read about <a href="https://www.ncetm.org.uk/resources/34332">one teacher's experience</a> of using this task in the classroom. </div> <h3>Key questions</h3> <ul> <li>You are being asked to sketch a family of graphs. What makes this a family?</li> <li>What is the same and what is different about the equations $y=x^2$ and $y=-(x-4)^2$?</li> <li>How might these similarities and differences relate to the way they look and their positions on the axes?</li> <li>Can you convince us that the rules you have found will work with graphs of other functions.</li> </ul> <h3>Possible extension</h3> <a href="http://nrich.maths.org/public/viewer.php?obj_id=777&part=">More Parabolic Patterns</a> and <a href="http://nrich.maths.org/public/viewer.php?obj_id=791&part=">Parabolas again</a> offer similar pictures to reproduce. <a href="http://nrich.maths.org/public/viewer.php?obj_id=802&part=">Cubics</a> uses graphs of cubic functions, and <a href="http://nrich.maths.org/public/viewer.php?obj_id=784&part=">Ellipses</a> gives the opportunity to investigate the equation of an ellipse. <h3>Possible Support</h3> Learners could begin by investigating translation of straight lines and look at how the equations change. </mdoxml> <mdoxml version="1.0">Try sketching the graph of $y=x^2$ on paper. What would you expect the graph of $y=-x^2$ to look like? What is the effect of the minus sign? Is this one of the graphs in the picture? What would you expect the graph of $y=(x-4)^2$ to look like? How would you expect the graph of $y=x^2$ to be transformed to give the graph of $y=(x-4)^2$? What about $y=-(x-4)^2$? Draw the graphs of these functions using graph drawing software or a graphics calculator if you have access to one or the other. Were your predictions right? What have you learnt from this example about reflections and translations of graphs and the corresponding equations of the functions? Now experiment with drawing the graphs of other functions and see if you can find the equations for all the graphs in the picture. </mdoxml> 2 3 0 0 0 1 0 Parabolic Patterns The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs. Advanced Algebra Transformation of functions Sequences, Functions and Graphs Graphs Sequences, Functions and Graphs Quadratic functions Coordinates and Coordinate Geometry Parabola Information and Communications Technology Graph plotters