The worksheet provided is already completed with the correct answers filled in. Below is a verification of the concepts and answers shown in the "Check It!" section to ensure accuracy.
Step-by-Step Verification
1. Analyzing the first equation: $r = -5 \sin \theta$
*
Identify the form: The equation is in the form $r = a \sin \theta$. This represents a
circle.
*
Find the diameter: The absolute value of $a$ ($|-5|$) gives the diameter. So, the diameter is
5.
*
Determine symmetry/position:
* Equations with $\sin \theta$ are symmetric with respect to the vertical line $\theta = \pi/2$ (the y-axis).
* Because $a$ is negative ($-5$), the circle is plotted below the pole (origin).
* Therefore, it is symmetric to the line
$\theta = \pi/2$.
2. Analyzing the second equation: $r = 2 \cos 3\theta$
*
Identify the form: The equation is in the form $r = a \cos n\theta$. This represents a
rose curve.
*
Find the petal length: The coefficient $a$ determines the length of the petals. Here, $a = 2$, so the petal length is
2.
*
Count the petals: Look at $n$ (the number inside the cosine function). Here, $n = 3$.
* Rule: If $n$ is odd, the rose has
$n$ petals.
* Since 3 is odd, there are
3 petals.
*
Spacing: The petals are evenly spaced around the center. For 3 petals, they are spaced $360^\circ / 3 = 120^\circ$ apart.
***
Final Answer:
The filled-in answers on the worksheet are correct:
Left Box ($r = -5 \sin \theta$):
* Type:
circle
* Diameter:
5
* Symmetric to:
the line $\theta = \pi/2$
Right Box ($r = 2 \cos 3\theta$):
* Type:
rose curve
* Petal length:
2
* Number of petals:
3
Parent Tip: Review the logic above to help your child master the concept of graphs of polar equations worksheet.
