Polar Equations

r= a+b cos(kΘ)

We are going to investigate the “n-leaf rose.”

a=1, b=1, k is varied

Here we see as k increases the number of petals or “leaves” increases.

The value k is the number of petals the function has.

We also see that since 1 is being added to 1cos(kΘ), the petals intercept both axes at 2.

In this instance, 1+1cos(kΘ) extends out to the right.

Now, let’s see what happens when a varies.

a is varied, b=1, k=1

We see that when k=1 the number of petals stays at one. As a increases, the size of the petal increases.

The function intercepts the axes at (k+1) in this case.

Now, let’s see what happens when b varies.

a=1, b varies, k=1

Once again, since k=1 the number of petals stays at one. As b increases the sizes of the petal increases, but only to the right.

In this instance the function intercepts the horizontal axis at k=1.

We have seen how the values of a, b, and k alter the n-leaf rose, but what if we do not have any a value at all?

r= b cos(kΘ)

K is varied.

Here we see the graph the same as the first graph we investigate, only smaller.

The size of the petals is scaled down by 1, since 1 is not being added to the function.

r= a+b sin(kΘ)

We’ve investigated how different values of a, b, and k alter the graph of the function.

Now lets investigate what happens when we use the sine function rather than the cosine function.

a=1, b=1, k is varied

We see here that the graph is the same as when using cosine, only rotated counter-clockwise 90 degrees.

The sine graph is similar to the cosine graph in all other ways.

The values of a, b, and varying k affect the graph in the same ways, as well.

r= b sin(kΘ)

Once again, we that the sine graph is very similar to the cosine graph when the value of a is taken away.

The size of the petals is scaled down by 1, since 1 is not being added to the function.

Overall, we have seen how different values of a, b, and k have affected both the sine and cosine graphs of the n-leaf rose.