Question

You are to create an original polar graph that displays a picture or an interesting design. Your graph must include, but is not restricted to the following: 1- circle (not cantered at the pole) a 2- rose 3- a limacon 4- Lemniscate 5- line (horizontal , vertical and slanted) 6- Cardioid Your finished product must include: • Printed polar graph (highlight required equations that match graphs) • Enhanced polar graph (shading, colour, or added details) with printed inspirational image or written inspiration • Required polar equations that define your polar graph • All points of intersections (for 2 equations) that have been determined graphically (label points) Written as polar coordinates • All points of intersections (for same 2 equations) that have been determined algebraically

Answer 1

1) Rose:

Rose curve equations have two forms: r = a cos(nθ) and r = a sin(nθ) where a ≠ 0 and n is a positive integers. Petals have length determined by a. If n is odd, the number of petals is n. However, if n is even, the number of petals is 2n.

The equation of a rose has two forms which are

`\begin{gathered} r=a\cos(n\theta) \\ r=asin(n\theta) \end{gathered}`

The graph of a polar rose is given below as

2)Limacon:

The polar equation is in the form of a limaçon, r = a – b cos θ. Since the equation passes the test for symmetry to the polar axis, we only need to evaluate the equation over the interval [0, π] and then reflect the graph about the polar axis.

`=a-bcos\theta`

3)A circle: