Problem 73 - Intermediate Algebra

Question

Write each equation in standard form, if it is not already so, and graph it. If the graph is a circle, give the coordinates of its center and its radius. If the graph is a parabola, give the coordinates of its vertex. $$ x=-\frac{1}{4} y^{2} $$

Step-by-Step Solution

Answer

The equation is a parabola with vertex at (0, 0) and opens to the left.

Step 1: Identify the Form of the Equation

The given equation is \( x = -\frac{1}{4} y^2 \). This equation is not in standard form yet for any conic sections such as a circle or a parabola. Let's rearrange and analyze it to see what it represents.

Step 2: Rearrange into Standard Form

This equation can be rewritten by comparing it to the standard form of a parabola. The general form is \( y^2 = 4px \). Here, rearrange the equation to this form: \( y^2 = -4x \). Thus, it is in the standard form of a parabola that opens left.

Step 3: Determine Characteristics of the Parabola

For the equation \( y^2 = -4x \), the vertex of the parabola is at the origin, \((0, 0)\). Since \(-4 = 4p\), solving gives \( p = -1 \). This tells us that the parabola opens to the left with the focus at \((-1, 0)\).

Step 4: Graph the Parabola

To graph the parabola, plot the vertex at \((0, 0)\). The parabola opens to the left. As \( p = -1 \), the distance from the vertex to the focus point along the x-axis is 1 unit in the negative x direction. The directrix is the vertical line \( x = 1 \).

Key Concepts

Parabola

A parabola is a unique type of conic section that is symmetrical and has a distinct U-shape. It is one of the simplest curves you can encounter in algebra and appears frequently in various mathematical contexts.

Parabolas are defined as the set of all points that are equidistant from a single point called the "focus," and a line called the "directrix." This specific property gives parabolas their characteristic shape. The axis of symmetry is a line that runs through the vertex and splits the parabola into two mirror-image halves.

Parabolas can open upwards, downwards, to the left, or to the right, depending on the equation used. Understanding the orientation and properties of a parabola is essential for graphing it correctly.

Standard Form of an Equation

The standard form for conic sections helps in simplifying and identifying the properties of the geometric figure in question. When dealing with parabolas, the standard form of the equation is crucial for identifying the orientation and position of the curve.

The standard form of a parabola that opens either to the left or right is given by:

• \( y^2 = 4px \)

For parabolas that open upwards or downwards, the standard form is:

• \( x^2 = 4py \)

In these equations, \( p \), a constant, indicates:

• The distance from the vertex to the focus.
• The direction in which the parabola opens, as determined by the sign of \( p \).

Rearranging an equation into one of these standard forms allows us to identify these key features more easily, which is crucial for graphing the parabola accurately.

Vertex of a Parabola

The vertex of a parabola is a critical point, as it is the point where the parabola changes direction. Understanding how to locate and interpret the vertex is fundamental in graphing and analyzing parabolas.

In standard form, when a parabola is expressed like \( y^2 = 4px \), the vertex is located at the origin,

• \((0,0)\)

For more complex equations, you might see translations that shift the vertex to a different position, such as

• \((h, k)\)

These translations from

• \((h,k)\)

shift the parabola without altering its intrinsic properties like the direction it opens. The vertex provides valuable information, such as the axis of symmetry of the parabola. It also helps in setting a foundation for determining other elements, like the focus and directrix of the parabola.