Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews February 8, 2023February 1, 2023

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Payment Details

Payment Summary

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

A typical polar equation is in the form r = f (θ), where f is some function ( of
θ). θ is the independent variable, and r is the dependent variable. The graph of a polar equation is the collection of all points that have at least
one set of polar coordinates that satisfy the equation (remember that a point has more
than one set of polar coordinates). Polar equations can be graphed by plotting points, and
ultimately, this is the best way to do it. But there are a number of shortcuts that are
useful for graphing polar equations.

Symmetry is an important property of any graph. Like functions are either odd, even, or
neither, based on their symmetry properties, graphs of polar equations can be symmetric
with respect to either the polar axis, the pole, or the line θ = , or none
of these. Knowing whether a graph is symmetric in any way simplifies the graphing
process.

If in the polar equation, (r, θ) can be replaced by (r, - θ)or(- r, Π - θ),
the graph is symmetric with respect to the polar axis. If in the polar equation, (r, θ) can be replaced by (- r, θ)or(r, Π + θ), the graph is symmetric with
respect to the pole. If in the polar equation, (r, θ) can be replaced by (r, Π - θ)or(- r, - θ), the graph is symmetric with respect to the line θ = . These rules are true, of course, but their converses are not. The graph of
a polar equation can be symmetric with respect to one of these axes (or the pole) and not
satisfy any of the test equations. These rules are only used to help sketch a graph.

Finding the maximum absolute value of r and the θ values for which r = 0 is
also a useful technique in sketching and analyzing the graph of a polar equation. If for
some θ, r = 0, the graph intersects the pole.

One final technique for sketching and analyzing the graph of a polar equation is finding
the intercepts of the graph; that is, where it intersects the lines θ = 0 and θ = . These lines correspond to the x and y axes in the rectangular
coordinate system. Let's examine a polar equation and sketch and analyze it.

r = 2sin(θ). It is not uncommon for a polar equation to contain a trigonometric
function, like this one. Performing the symmetry tests, it is found that, because
sin(θ) = sin(Π - θ), the graph is symmetric with respect to the line θ = . This means we only need to plot values of θ for [0,]and[, 2Π), or[, Π]and (Π,]. If we can
plot the graph for values of θ in either of these two sets of intervals, we can use the
symmetry of the graph to sketch it for the other values of θ. The maximum
absolute value of r occurs when sin(θ) = 1or - 1; therefore, θ = ,, and r = 2, - 2, respectively. Both of these ordered
pairs specify the same point. r = 0 when sin(θ) = 0, which is true for θ = 0, Π. Finally, evaluating the equation at θ = 0,, we find the
intercepts are at (0, 0)and (2,).

At this point, we plot some sample points of the equation, along with the maximum and
zero values of r and the intercepts. Using the symmety of the graph, we find that the
graph looks like this:

We also find that the entire graph is drawn using the values of θ from 0toΠ.

There are a few well-known names for special kinds of graphs which are more simply
defined by polar equations than rectangular.

A limacon is a curve with the equation r = a + b sin(θ)orr = a + b cos(θ),
where a, b≠ 0. Below is the limacon r = 2 + 3 cos(θ).

A rose curve is a curve with the equation r = a sin(nθ) or r = a cos(nθ),
where n is an integer. Each loop in a rose curve is called a petal. The number of petals
in a given curve is n if n is odd, and 2n if n is even. The length of each petal is
a. Below is the rose curve r = 3 sin(2θ).

Two common kinds of spirals are called spirals of Archimedes and logarithmic spirals. A
spiral of Arhcimedes is of the form r = aθ + b, and a logarithmic spiral is of the
form r = ab^{θ}. They are pictured below.

The common circle with its center at the pole comes from the equation r = c, where c
is a constant. A circle that intersects the pole once comes from the equation r = a sin(θ) or r = a cos(θ), with a diameter of a. The example explained
earlier is a circle that intersected the origin once.

Because polar equations often contain trigonometric functions, their graphs often repeat
themselves (the trigonometric functions are periodic). In such cases, the entire graph
may be traced within a small interval of values of θ. Usually, the period of the
given trigonometric function is sufficient to trace the entire graph, but sometimes it is not.

The safest way to graph a polar equation is to plot points until you have a feel for what
the graph looks like. All of the hints in this section are only aids in sketching a graph of a
polar equation.