Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. \(y^{2}=4-x^{2}\)

Step-by-Step Solution

Answer

The equation forms a circle, not a parabola.

Step 1: Review the Given Equation

The equation provided is \(y^2 = 4 - x^2\). Compare this to the general form of a conic section. For a parabola, the standard form should look like \((x-h)^2 = 4p(y-k)\) or \((y-k)^2 = 4p(x-h)\). We need to determine if it can fit one of these forms.

Step 2: Rearrange the Equation

Attempt to rearrange or manipulate the equation to see if it can be expressed in a parabolic form. Begin by rewriting the given equation: \[ y^2 + x^2 = 4 \]This form is closer to that of a circle rather than a parabola.

Step 3: Identify the Type of Conic Section

Compare \(y^2 + x^2 = 4\) to the standard conic sections. A circular equation has the form \(x^2 + y^2 = r^2\) for a radius \(r\). Here, the radius is 2 (since \(r^2 = 4\)), indicating it forms a circle, not a parabola.

Key Concepts

Parabola

A parabola is a U-shaped curve that can open in various directions, such as up, down, left, or right. The defining feature of a parabola is its symmetry across a line, known as the axis of symmetry. In mathematical terms, a parabola is the set of all points that are equidistant from a fixed point called the "focus" and a line named the "directrix." This leads to its characteristic equation form:

For vertical parabolas, the equation follows the form \((x-h)^2 = 4p(y-k)\), while for horizontal parabolas, it follows \((y-k)^2 = 4p(x-h)\). In these equations, \(h\) and \(k\) are the coordinates of the parabola's vertex, and \(p\) represents the distance from the vertex to the focus or directrix.

• Vertical parabola: opens upward or downward depending on the sign of \(p\).
• Horizontal parabola: opens leftward or rightward depending on the sign of \(p\).

This exercise's equation, upon inspection, was found not to fit these forms, indicating it’s not a parabola.

Standard Form

Understanding the standard form of different conic sections is crucial. For parabolas, the standard form is either \((x-h)^2 = 4p(y-k)\) or \((y-k)^2 = 4p(x-h)\). Comparatively, for circles, the standard form is \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle.

Moving an equation into standard form often involves manipulating the given equation through algebraic operations such as completing the square or rearranging terms. For the parabola, standard form reveals important information: its vertex \( (h, k) \) and the value \(p\), which tells how far the parabola opens from the vertex.

It's important to compare your equation against these forms. If an equation fits neither the parabola nor circle standard forms, further investigation is required to identify the correct conic section.

Circle

A circle is a round shape that is defined in geometry as the set of all points equidistant from a central point. The equation for a circle is typically \( x^2 + y^2 = r^2 \), where \( r \) is the radius. This equation means every point \( (x, y) \) lies \( r \) units away from the center at the origin.

In the given exercise, the manipulation of the equation \( y^2 = 4 - x^2 \) into \( x^2 + y^2 = 4 \) suggests it represents a circle with a radius of 2. Here, simply matching the rearranged equation with \( x^2 + y^2 = r^2 \) confirms that the conic section is a circle rather than a parabola.

• Radius (\