Problem 25 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The parabola \(y^2 = -\frac{1}{3}x\) opens left with vertex at (0,0).

Step 1: Identify and Rearrange the Equation

The equation given is \(y^2 = -\frac{1}{3}x\). This can be identified as a parabola in the form \(y^2 = 4px\), where \(4p = -\frac{1}{3}\). Rearranging gives us \(p = -\frac{1}{12}\).

Step 2: Determine the Direction of the Parabola

Since \(p = -\frac{1}{12}\) is negative, the parabola will open to the left on the coordinate plane.

Step 3: Find the Vertex of the Parabola

The vertex of a parabola in the form \(y^2 = 4px\) is at the origin, (0, 0), because no additional terms are added to the equation. Thus, the vertex is \((0, 0)\).

Step 4: Sketch the Parabola

With the vertex at (0,0) and knowing that the parabola opens to the left, plot several points by substituting values for \(y\) to find corresponding \(x\) values. For example, using \(y = 0, 1, -1\), we find \(x = 0, -3, -3\) respectively. Plot these points and the symmetric ones and draw the curve.

Step 5: Use a Graphing Device

Enter the equation \(y^2 = -\frac{1}{3}x\) into a graphing calculator or online graphing tool. Adjust the window settings to include negative \(x\)-values prominently since the parabola opens to the left. Review the generated graph to confirm it matches the manually sketched graph.

Key Concepts

graphing calculator

A graphing calculator is a powerful tool that helps you visually understand mathematical concepts, like parabolas, through graphs. When you input an equation, such as \( y^2 = -\frac{1}{3}x \), the graphing calculator plots the curve on a coordinate plane for you. This is particularly helpful for more complex equations or when you want to quickly visualize the behavior of a graph without manual calculations.

To use a graphing calculator effectively:

• Ensure you input the equation correctly. Double-check for any missing or extra characters.
• Adjust the viewing window. For a parabola like our example, you need to adjust for a sufficient range of \( x \) and \( y \) values, especially focusing on negative \( x \) values as the parabola opens left.
• Use features like zoom or trace to explore specific parts of the graph.
• Verify your manual sketch by comparing it with the graph generated by the calculator.

These steps help you check your work and understand geometric changes visually.

coordinate plane

The coordinate plane is a two-dimensional surface where you can plot points, lines, and curves using an \( x \) and \( y \) axis. Understanding how to use the coordinate plane is essential for graphing parabolas and other functions.

Some things to keep in mind:

• The \( x \)-axis runs horizontally, while the \( y \)-axis runs vertically.
• The point where these axes intersect is called the origin, marked as \((0,0)\).
• Positive values are to the right on the \( x \)-axis and upward on the \( y \)-axis, while negative values are to the left and downward, respectively.

In our parabola's case, you start plotting from the vertex at the origin and consider both axes' directions as you plot additional points like \((y = 0, x = 0)\), \((y = 1, x = -3)\), ensuring they align with the equation's predicted path. This visual framework helps you appreciate the function's structure and direction.

vertex of a parabola

The vertex of a parabola is its highest or lowest point, or in cases like our example, the point from which it curves outwards. For parabolas like \( y^2 = 4px \), the vertex is often at \( (0,0) \). This provides a starting reference point for graphing.

When identifying the vertex:

• Check if the equation is in standard form to easily locate the vertex. For our equation \( y^2 = -\frac{1}{3}x \), the vertex can be read directly as \((0,0)\).
• Remember, the vertex informs the parabola's orientation and directional opening.
• For horizontal parabolas like ours, extend from the vertex to the left or right depending on the equation.

Correctly identifying the vertex is crucial as it guides subsequent points for plotting and ensures the graph's accuracy. It serves as an anchor for drawing and understanding the parabola's shape and orientation.