Problem 27 - Precalculus: Mathematics for Calculus

Question

Use a graphing device to graph the parabola. $$y^{2}=-\frac{1}{3} x$$

Step-by-Step Solution

Answer

Graph the parabola opening to the left with vertex at (0,0).

Step 1: Writing in Standard Form

The equation given is \(y^2 = -\frac{1}{3}x\). This is already in a form suitable for graphing, where \(y^2 = -4p(x - h)\) and \((h, k)\) is the vertex of the parabola. Here, the vertex is at \((0, 0)\) and \(p = \frac{1}{12}\).

Step 2: Determine the Direction

Since the coefficient of \(x\) in \(y^2 = -\frac{1}{3}x\) is negative, the parabola opens to the left. This means the graph will be a horizontal parabola facing left.

Step 3: Identifying the Vertex

The vertex of the parabola \(y^2 = -\frac{1}{3}x\) is located at the origin, \((0,0)\). This means this point will be the center from which the parabola is symmetric.

Step 4: Calculate a Few Points

Select some values of \(y\) and calculate corresponding \(x\) values. For example, if \(y = 1\), then \(1^2 = -\frac{1}{3}x\) gives \(x = -3\). Similarly calculate for \(y = -1\), resulting in \(x = -3\). These points help in sketching the parabola.

Step 5: Graph the Parabola

Use the vertex \((0,0)\) and the calculated points \((3, 1)\) and \((3, -1)\) to sketch the parabola. Since the parabola opens to the left, it will be symmetric about the x-axis and will expand leftward as \(y\) values increase or decrease.

Key Concepts

Vertex Form

The concept of the vertex form is crucial in understanding parabolas. The vertex form of a parabola is typically expressed as \(y = a(x-h)^2 + k\) for vertical parabolas, where \((h, k)\) is the vertex. In the case of horizontal parabolas, as seen in the equation \(y^2 = -\frac{1}{3}x\), we need to use the form \(y^2 = 4p(x - h)\). The vertex, \((h, k)\), plays a central role as it represents the point from which the parabola is symmetric with respect to its axis.
In our example, the vertex is at \((0, 0)\), making it simple to identify the center of symmetry.

• The vertex form helps not only in graphing but also in quickly determining the vertex and the axis of symmetry.
• This formulation aids in understanding how the parabola might change when the coefficients or vertex \((h, k)\) vary.

Horizontal Parabola

Horizontal parabolas are parabolas that open sideways from their vertex. They can face either left or right, depending on the direction dictated by the coefficient of \(x\) in their equation. When \(x\) has a negative coefficient, like in \(y^2 = -\frac{1}{3}x\), the parabola opens to the left.
In contrast, a positive \(x\) coefficient opens the parabola to the right.
Key features of a horizontal parabola include:

• The shape is symmetrical with respect to the y-axis.
• The vertex \((h, k)\) is at the center, guiding the structure's symmetric nature.
• The axis of symmetry is perpendicular to the direction the parabola opens.

Understanding these elements helps you effectively sketch a horizontal parabola by determining its broad direction and general structure.

Parabola Symmetry

Symmetry in parabolas refers to the mirror-like format of the graph across the axis. For a horizontal parabola, its axis of symmetry lies vertically. The axis passes through the vertex, \((h, k)\), and is a line that splits the parabola into two mirror-image halves.
The equation \(y^2 = -\frac{1}{3}x\) shows that the parabola is symmetric about the x-axis, considering \(y\) can take on both positive and negative values with the same \(x\).

• This characteristic makes it easier to sketch the parabola once a few points are determined on one side of the axis.
• Recognizing symmetry allows for more efficient graphing because each positive \(y\) value’s result can be mirrored on the negative side.

Essentially, this symmetry definition helps determine additional points for plotting the parabola without repetitive calculations.

Parabola Direction

The direction in which a parabola opens is determined by the sign and value of the coefficient in front of \(x\) or \(y\) in its equation. In the case of our exercise, the equation \(y^2 = -\frac{1}{3}x\) has a negative coefficient for \(x\), indicating a left-opening parabola.
The main points about parabola direction include:

• For horizontal parabolas, a negative coefficient makes the parabola face left; a positive one faces right.
• This direction influences the shape and graphing strategy, as it defines the orientation of the curve relative to the vertex.
• As you plot points to sketch the graph, keep the opening direction in mind to maintain accuracy.

Understanding this aspect helps you anticipate and check the correctness of your graph as you draw out the parabola.