Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational
symmetry. Do graphs of all cubics have rotational symmetry?
In this 'mesh' of sine graphs, one of the graphs is the graph of
the sine function. Find the equations of the other graphs to
reproduce the pattern.
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.
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
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
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 $a=3b$.