Question

Sketch a graph of the polar equation and identify any symmetry. $$ r^{2}=4 \sin \theta $$

Short answer

The graph is a circle centered at (0,2) with radius 2; it has y-axis symmetry.

Step-by-step solution

Convert polar to rectangular

To start examining the polar equation \( r^2 = 4 \sin \theta \), convert it to a rectangular form. We know that \( r^2 = x^2 + y^2 \) and \( \sin \theta = \frac{y}{r} \). Substituting these into the given equation leads to \( x^2 + y^2 = 4 \frac{y}{r} \), which simplifies to \( x^2 + y^2 = 4y/r \), and finally to \( x^2 + y^2 = 4y \).

Use algebra to simplify

Rearrange the rectangular equation \( x^2 + y^2 = 4y \) to see if it resembles a known form. Completing the square for the \( y \)-terms gives \( x^2 + (y-2)^2 = 4 \), which represents a circle centered at \((0, 2)\) with radius 2.

Determine symmetry

Next, check for symmetries in the polar equation. Substituting \( \theta + \pi \) and checking if \( r \) remains unchanged can indicate symmetry. If we replace \( \theta \) with \( 2\pi - \theta \), because \( \sin(-\theta) = -\sin(\theta) \), we find \( r(-\theta) = r(\theta) \) does hold, showing symmetry about the line \( \theta = \frac{\pi}{2} \).

Sketch the graph

Use the equation \( x^2 + (y-2)^2 = 4 \) to sketch the graph. The center of the circle is at \((0, 2)\), and it has a radius of 2. This graph will be entirely above the x-axis, symmetric about the line \( y = 2 \) in the rectangular coordinate system.

Key concepts

Sketching Graphs

Plotting graphs from polar equations can be an exciting exploration. To sketch a graph, you first need to understand the polar equation you are dealing with. In our case, we have the equation \( r^2 = 4 \sin \theta \). Turning this into a graph involves a few steps.

• Convert from polar to rectangular form: Use the relationships \( r^2 = x^2 + y^2 \) and \( \sin \theta = \frac{y}{r} \).
• Algebraically simplify it: In this exercise, it forms a circle \( x^2 + (y-2)^2 = 4 \).
• Learn its basic geometric form: Recognize that this expression represents a standard geometric shape - a circle centered at \((0, 2)\) with a radius of 2.

Plotting such a graph allows you to visualize the curve, and can help in understanding its different properties such as symmetry and boundary. Remember, polar graphs can sometimes behave differently than what we expect in rectangular coordinates.

Symmetry in Equations

Understanding symmetry aids in simplifying and anticipating the behavior of graphs. Symmetry is essentially when one part of a graph mirrors another, and it can significantly assist in sketching graphs more effectively.

To determine symmetry for the polar equation \( r^2 = 4 \sin \theta \), you would perform a series of checks:

• Test angular symmetry, \( \theta \rightarrow \theta + \pi \), which often checks for rotational symmetry. Though not applicable here, it is a key method.
• Check the reflectional symmetry: Replace \( \theta \) with \( 2\pi - \theta \) in the equation. If the equation remains unchanged, the graph is symmetric with respect to the polar axis.
• In our particular problem, substituting yields that reflection across the line \( \theta = \frac{\pi}{2} \) shows valid symmetry.

Symmetry not only makes graphing easier but it also helps in understanding curves' nature and how they interact with different axes.

Rectangular Coordinates Conversion

Converting between polar and rectangular coordinates is a foundational skill that opens the door to analyzing equations in different formats.

In this transformation, you change expressions involving \( r \) and \( \theta \) into forms using \( x \) and \( y \). For instance, to convert the given polar equation \( r^2 = 4 \sin \theta \), understand these key points:

• The relationship \( r^2 = x^2 + y^2 \) allows you to start the conversion by substituting directly.
• Use \( \sin \theta = \frac{y}{r} \) to replace trigonometric functions with algebraic expressions.
• Simplify into a standard form, such as \( x^2 + y^2 = 4y \), which further reduces to \( x^2 + (y - 2)^2 = 4 \).

This circle equation can be easier to understand and visualize. Knowing how to switch between coordinate systems enables you to apply the best approach for solving, graphing, or interpreting geometrical problems.