Question

Convert the rectangular equation to polar form and sketch its graph. $$ y^{2}=4 x $$

Short answer

The polar form is \( r = \frac{4 \cos \theta}{\sin^2 \theta} \).

Step-by-step solution

Express the Cartesian Coordinates in Polar Coordinates

The rectangular equation given is \( y^2 = 4x \). In polar coordinates, we have the relationships \( x = r \cos \theta \) and \( y = r \sin \theta \). These relationships will help us convert the given equation into polar form.

Substitute the Polar Coordinates

Substitute \( x = r \cos \theta \) and \( y = r \sin \theta \) into the equation \( y^2 = 4x \). This gives us \((r \sin \theta)^2 = 4 (r \cos \theta) \).

Simplify the Polar Equation

Simplifying \( (r \sin \theta)^2 = 4 (r \cos \theta) \) leads to \( r^2 \sin^2 \theta = 4r \cos \theta \). Assuming \( r eq 0 \), divide both sides by \( r \), yielding \( r \sin^2 \theta = 4 \cos \theta \).

Solve for \( r \)

Rearrange the equation \( r \sin^2 \theta = 4 \cos \theta \) to solve for \( r \). We get \( r = \frac{4 \cos \theta}{\sin^2 \theta} \). This is the polar form of the given rectangular equation.

Sketch the Graph

The polar equation \( r = \frac{4 \cos \theta}{\sin^2 \theta} \) represents a parabola that opens to the right. The graph is similar to half of a horizontal parabola, symmetric with respect to the x-axis. As \( \theta \) approaches 0 or \( \pi \), \( r \) approaches infinity, indicating the parabola extends infinitely to the right and left, respectively.

Key concepts

Rectangular Equation

A rectangular equation, sometimes known as a Cartesian equation, is an equation that defines a curve in the x-y coordinate plane using Cartesian coordinates. In our example, the rectangular equation is given as \( y^2 = 4x \). Understanding rectangular equations is crucial because they often describe familiar geometric shapes, such as lines, circles, ellipses, and in this instance, a parabola. They use familiar \( x \) and \( y \) coordinates to express relationships between points in the plane.
For this equation, the "standard form" shows a parabola opening horizontally. The \( y^2 \) term implies symmetry about the x-axis, and the expression \( 4x \) suggests the parabola's directionality to the right. This is useful before converting to polar coordinates since it provides a sense of the curve's behavior and graph.

Parabola in Polar Form

Polar coordinates offer a different way to represent curves through points defined by a distance from a fixed point (the pole) and an angle from a fixed direction (typically the positive x-axis). The trick is in converting equations from one form to another, such as parabolas in rectangular equations to polar coordinates.
Our original problem instructed the conversion of the rectangular equation \( y^2 = 4x \) into polar form. This transformation yields the polar equation:\[ r = \frac{4 \cos \theta}{\sin^2 \theta}\]This equation represents the same parabola but expressed in terms of distance \( r \) and angle \( \theta \).

• Notice that polar equations can provide clarity on features like direction and extending behavior, such as a parabola's opening direction and its infinite extension.
• In polar form, the angle \( \theta \) greatly influences \( r \), showing how the parabola extends infinitely as \( r \) approaches infinity when \( \theta \) approaches 0 or \( \pi \).

Polar Equation Conversion

To convert a rectangular equation to a polar equation involves using the essence of both coordinate systems. Specifically, knowing that \( x = r \cos \theta \) and \( y = r \sin \theta \) allows for this conversion.
This process is fairly systematic:

• Start by substituting \( x \) and \( y \) from the rectangular equation with their polar equivalents.
• With this, \( y^2 = 4x \) becomes \((r \sin \theta)^2 = 4 (r \cos \theta)\).
• Next, simplify and rearrange the equation so \( r \) can be solved as a function of \( \theta \).

The result of such conversions, as seen, can open new insights, like seeing infinite extensions of curves or symmetry not immediately obvious in their rectangular forms. Utilization of these conversions can aid in deeply grasping the geometry behind equations.