Cubics

802 /www/nrich/html/content/01/10/six5/ 1 2011-02-01T00:00:01 <mdoxml version="1.0"> <mdo:image alt="graph." src="cubics.gif"></mdo:image> This pattern of graphs is the creation of Ali Abu-Hijleh, from Riccarton High School, Christchurch, New Zealand. Two of the equations are: $$y=(x+6)^3-2$$ $$y=-(x-9)^3+3$$ Find the equations of the other 12 graphs in this pattern. You could use a graphic calculator or graphing software. For example, you can download Graphmatica for free from <a href="http://nrich.maths.org/downloads/graphmatica.zip">here</a> and it comes with a good help file. Draw your own pattern of graphs. </mdoxml> <mdoxml version="1.0"> Many of you have sent in absolutely stunning graph patterns that you have created using a single graph and transformations of it. You have discovered how to translate the graph and how to stretch it in different directions by changing the equation of the graph accordingly. <mdo:image height="287" width="500" src="cubics.gif" alt="graph."></mdo:image> This is Ali Abu-Hijleh's pattern. We gave you two of the equations: $$y = (x+6)^3 - 2$$ $$y=-(x-9)^3+3$$and here are the equations of the other curves: $$y=x^3$$ $$y=-x^3$$ $$y=-(x+9)^3-3$$ $$y=(x+9)^3-3$$ $$y=-(x+6)^3-2$$ $$y=(x+6)^3-2$$ $$y=-(x+3)^3-1$$ $$y=(x+3)^3-1$$ $$y=-(x-3)^3+1$$ $$y=(x-3)^3+1$$ $$y=-(x-6)^3+2$$ $$y=(x-6)^3+2$$ $$y=-(x-9)^3+3$$ $$y=(x-9)^3+3$$Well done Hsiu Chen; Claire from the Mount School, York; Malcolm and Charles from Madras College, St Andrew's and Andrei from School No. 205, Bucharest, Romania. All these students gave the equations with very good explanations of their methods as follows: I started from the graph of the function: $$y=x^3$$ then I identified it on the pattern. I saw that the reflection of this graph, either in the $x$-axis or in the $y$ axis, gives: $$y=-x^3%$$I displaced these graphs along the $x$ and $y$ axes. I identified the scale by looking at the two equations given in the problem. The scale in the drawing is: $x$ between $-12$ and $12$, and $y$ between $-6$ and $6$. All the equations are of the form $-y = \pm (x\pm a)^3\pm b $ where adding and subtracting $b$ translates the graph up and down parallel to the y-axis and changing $x$ to $x\pm a$ translates the graph parallel to the x-axis. It happens that in this example we always have $a=3b$. </mdoxml> <mdoxml version="1.0"> This is another inverse challenge. You are given the graphs and you have to use mathematical reasoning to find their equations.</mdoxml> <mdoxml version="1.0">Sketch the graph of the cubic $y=x^3$ which has a point of inflection at the origin. Think about the effects of transforming this graph by translating it parallel to the axes and reflecting it. </mdoxml> 2 3 0 0 0 1 1 Cubics Knowing two of the equations find the equations of the 12 graphs of cubic functions making this pattern. Information and Communications Technology Graph plotters Sequences, Functions and Graphs Graphs Sequences, Functions and Graphs Cubic functions Advanced Algebra Transformation of functions