Polar flower
This polar equation is a quadratic. Plot the graph given by each factor to draw the flower.
Problem
Factorise the equation $$ r^2 - r + {\textstyle{1\over 4}} \sin^2 4\theta = 0$$ and hence sketch the graph given by this polar equation. What features do you notice and how do these arise?
Getting Started
The graph of $xy=0$ is the pair of lines $x=0$ and $y=0$. Similarly you will need to draw a pair of graphs here but this time in polar coordinates.
When you plot the point $(r,\theta)$ in polar coordinates, the length of the line segment joining this point to the origin is $r$ and the angle between this line segment and the positive direction of the $x$ axis is given by $\theta$.
Student Solutions
Elizabeth from the International School of Geneva, someone from Winchester College who gave no name and Andrei from Tudor Vianu National College, Bucharest, Romania all sent solutions to this problem.
Elizabeth factorised the expression as follows:
$$\eqalign{ r^2 - r + {\textstyle{1\over 4}} \sin^2 4\theta &= 0 \cr r^2 - r + {\textstyle{1\over 4}}(2(\sin 2\theta \cos 2\theta))^2 &= 0 \cr r^2 - r + \sin^2 2\theta \cos^2 2\theta &= 0 \cr (r - \sin^2 2\theta)(r- \cos^2 2\theta) &=0.}$$
and plotted the graphs of $r = \sin^2 {2\theta}$ and $r=\cos^2 2{\theta}$.
Andrei factorised the expression differently
$$(2r+\cos {4\theta} -1)(2r \cos {4\theta} - 1)$$
and plotted the graphs of $r = \textstyle{1\over 2}(1 + \cos 4\theta)$ and $r = \textstyle{1\over 2}(1 - \cos 4\theta)$ but you should be able to see that the results are equivalent.
Our friend from Winchester College used the difference of two squares to arrive at the same form of this result as Andrei and another method would be substitution in the formula for the solution of a quadratic equation.
Consider first the graph of $r- \sin^2 2\theta=0$ in polar coordinates where $r$ is the length of the line segment from the point from the origin and $\theta$ is the angle measured counter clockwise between this line segment and the positive $x$ axis. For points on this graph, as $\theta$ increases from 0 to ${\pi \over 4}$ we have $\sin^2 2\theta$ increases from 0 to 1. Between $\theta = {\pi\over 4}$ and ${\pi \over 2}$ the value of $r$ decreases from 1 to 0 so that the graph in the first quadrant is a 'petal' symmetrical about the line $\theta = {\pi \over 4}$.
Similarly the graph in the second quadrant is a 'petal' for $\theta $ between ${\pi \over 2}$ and $\pi$ where $r$ takes positive values corresponding to $\sin^2 \theta$. The graph in the third quadrant is a 'petal' for $\theta $ between $\pi $ and ${3\pi \over 2}$ and the graph in the fourth quadrant is a 'petal' for $\theta $ between ${3\pi \over 2}$ and $2\pi$.
Next consider the graph of $r- \cos^2 2\theta=0$. This will be of the same form but rotated by ${\pi \over 4}$ corresponding to the phase shift between the graphs of $\sin 2\theta$ and $\cos 2\theta$.
For points on this graph, as $\theta$ increases from 0 to ${\pi \over 4}$ we have $\cos^2 2\theta$ decreases from 1 to 0. Between $\theta = {\pi\over 4}$ and ${\pi \over 2}$ the value of $r$ increases from 0 to 1 and between ${\pi \over 2}$ and ${3\pi \over 4}$, as $r$ decreases from 1 to 0, the 'petal' symmetrical about the y-axis is completed.
The next petal, symmetrical about the negative x-axis, is drawn for $\theta $ between ${3\pi \over 4}$ and ${5\pi \over 4}$. The next petal, symmetrical about the negative y-axis, is drawn for $\theta $ between ${5\pi \over 4}$ and ${7\pi \over 4}$ and the remaining petal is completed for $\theta $ between ${7\pi \over 4}$ and $2\pi$.
Teachers' Resources
Why do this problem?
This problem provides students who have a good grasp of plotting simple functions using polar coordinates with the opportunity to extend their thinking and also produce a pleasing picture. It also reinforces thinking about the point of factorising, which can often be a mechanical process which is given little thought. The problem uses trigonometric identities as well, building fluency and the awareness of useful equivalent forms.
Possible approach
Students should have a good grounding in plotting simple polar graphs such as $r=\sin a\theta$ before attempting this problem.
A starting point might be to present the equation to students without any context. What do they think of it? Can they solve it? (What would that mean?) What else could they do with it? Would a graphical approach help? (How would that work?) The hope is that students will spot that the equation includes both $r$ and $\theta$, so it is not likely to have a single solution, that it is quadratic and perhaps factorisable, and that graphing polar curves works best when they are in the form $r=f(\theta)$ (or perhaps $r=k$ or $\theta=k$). If students do not make any of these observations, they might be guided by further questions such as "What sort of equation is it?" or "What sorts of polar equations do we know how to graph? Is it possible to get this equation into such a form?"
Once students have generated some ideas, they could work on manipulating the equation individually or in pairs. The teacher could circulate to ensure they are making some progress and provide hints if needed, perhaps suggesting the quadratic formula or trig identities if students are really stuck. There are many possible approaches to factorising the equation, so if time permits, students should be allowed to pursue their own ideas.
When students have factorised the equation, they should think about how that allows them to plot the graph. Ideally, they should attempt a sketch before turning to graphics calculators or graph drawing software for a more accurate plotting. In pairs or as a whole class, students should discuss the features of the graph (symmetry, maximum and minimum values, overlap, etc) and relate these to aspects of the equation.
Possible extension
Students can go on to find areas associated with the graph, such as the area of one "petal" (are they all the same?) or more challengingly, the area of the overlap between two petals. They could also try to produce a similar problem by combining two polar curves.
Possible support
Students can be guided to using a particular approach to factorising the quadratic, such as by introducing specific trig identities early on, using the quadratic formula and following with trig identities or completing the square and factorising using difference of two squares (and more trig identities!). They could also be given access to graphics calculators or graph drawing software from the outset.