Problem 25 - Algebra and Trigonometry

Question

\(23-28\) Use a graphing device to graph the parabola. $$ y^{2}=-\frac{1}{3} x $$

Step-by-Step Solution

Answer

Graph the parabola opening left with vertex at origin, using \( y^2 = -\frac{1}{3} x \).

Step 1: Identify the Parabola's Equation Type

The given equation is \( y^2 = -\frac{1}{3} x \). This is a sideways parabola, which means it opens leftwards because of the negative coefficient in front of \( x \).

Step 2: Understand the Graph's Orientation

In the context of \( y^2 = -4ax \), the standard form of a horizontal parabola, the equation \( y^2 = -\frac{1}{3} x \) means that \(-4a = -\frac{1}{3}\), thus \( a \) is positive and equal to \( \frac{1}{12} \). This confirms the parabola opens to the left.

Step 3: Determine the Vertex

Since the equation is \( y^2 = -\frac{1}{3}x \), the vertex of this parabola is at the origin \( (0, 0) \).

Step 4: Graph the Parabola Using a Graphing Device

Use a graphing device to input the equation \( y^2 = -\frac{1}{3}x \). Ensure that the graph correctly represents a parabola opening to the left with its vertex at the origin.

Key Concepts

sideways parabola

When talking about parabolas, a typical thought is the "U" or "n" shapes that open upwards or downwards. But, did you know they can also lie sideways? Sideways parabolas are paraboloids that open towards the left or right.
Their general form is expressed with squared terms involving the variable for the vertical axis, such as in the equation \(y^2 = -\frac{1}{3} x\).
The direction in which a sideways parabola opens is determined by the coefficient of the linear term.

• A negative coefficient, like in the equation \(y^2 = -\frac{1}{3} x\), signals that the parabola opens to the left.
• A positive coefficient would mean the parabola opens to the right.

A sideways parabola showcases a flipped understanding of parabolic curves but is just as essential in graphing applications.

vertex of a parabola

The vertex of a parabola is its turning point or the point where it is at its widest. For a sideways parabola, this concept remains crucial because the vertex still dictates the shape and positioning of the graph.
In standard form of a sideways parabola like \(y^2 = 4ax\), the vertex is generally found at the coordinates \((h, k)\).
However, if the equation is presented without shifts, such as \(y^2 = -\frac{1}{3} x\), the vertex lies right at the origin \((0, 0)\).

• The vertex provides a reference for graphing and can serve as a starting position for plotting other points of the parabola.

Understanding where the vertex lies helps bring clarity to the overall graphing process.

graphing parabolas

Graphing parabolas becomes a delight once you grasp the basics, especially for sideways parabolas. In the exercise, you are encouraged to use a graphing device, which allows us to visually see the embrace of the sideways parabola. Here’s how you could do it manually:

• Start by identifying the vertex. For \(y^2 = -\frac{1}{3}x\), it's at \((0,0)\).
• Next, consider the direction it opens. A negative coefficient indicates leftward.
• Plot additional points by substituting values for one variable and solving for the other.

This process not only helps with visual learning but also allows you to cross-check your work, ensuring that the characteristic of the sideways parabola is correctly depicted in its graphical form.

equation of a parabola

The equation of a parabola can take various forms, each revealing unique aspects about the parabola’s orientation and position. The sideways parabola, as in \(y^2 = -\frac{1}{3}x\), flips the typical equation on its head.

• Horizontal orientation: The squared term is \(y^2\), which suggests a horizontal opening.
• Coefficient significance: The value \(-\frac{1}{3}\) reflects the parabola's width and orientation. A smaller absolute value of this coefficient indicates a wider parabola, while a larger value narrows it.

The understanding of the equation helps uncover the geometry of the parabola, making it easier to predict and plot its graph. Being able to translate the algebraic form of the equation into a graphical representation is a handy skill in mathematics, showcasing the interconnectedness of different math areas.