5 Steps to a 5 AP Calculus BC 2019

130 STEP 4. Review the Knowledge You Need to Score High

(continued)

SHAPE TYPICAL EQUATION NOTES

Rose r=asin(nθ) Length of petal=a

r=acos(nθ)Ifnis odd,npetals.

Ifnis even, 2npetals.

Cardiod r=a±asinθ

r=a±acosθ

Limaçon r=a±bsinθ If

a

b

<1, limaçon has an inner loop.

r=a±bcosθ

Spirals r=aθ

r=a

√

θ

r=

a

θ

r=

a

√

θ

r=aebθ

Example 1

Classify each of the following equations according to the shape of its graph.

(a)r= 5 +7 cosθ, (b)r=

4

θ

, (c)r= 4 −4 sinθ.

The equation in (a) is a limaçon, and since

5

7

<1, it will have an inner loop. The equation

in (b) is a spiral. Equation (c) appears at first glance to be a limaçon; however, since the

coefficients are equal, it is a cardiod.

Example 2

Sketch the graph ofr=3 cos( 2 θ). The equationr=3 cos( 2 θ)is a polar rose with four petals

each 3 units long. Since 3 cos(0)=3, the tip of a petal sits at 3 on the polar axis.

Symmetry of Polar Graphs

A polar curve of the formr= f(θ) will be symmetric about the polar, or horizontal, axis if

f(θ)= f(−θ), symmetric about the lineθ=

π

2

if f(θ)= f(π−θ), and symmetric about

the pole iff(θ)= f(θ+π).