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Magazine Chapter 11: Problem 21
Use a graphing device to graph the parabola. $$y^{2}=-\frac{1}{3} x$$
Short Answer
Expert verified
Graph the parabola opening to the left with vertex at (0,0).
Step by step solution
01
Recognize the Equation Type
The given equation is \( y^2 = -\frac{1}{3}x \), which is in the form of a parabola. In general, if you have \( y^2 = kx \) or \( x^2 = ky \), it represents a parabola.
02
Determine the Orientation
Since the equation is of the form \( y^2 = -\frac{1}{3}x \), this indicates a horizontal parabola which opens to the left because of the negative sign before \( \frac{1}{3}x \).
03
Identify Key Features
Identify the vertex of the parabola and any other key points. Here, the vertex of the parabola is at the origin (0,0).
04
Graph the Parabola
Using a graphing device, first plot the vertex at (0,0). Next, draw the parabola opening to the left. Select values of \( y \), plug them into the equation, and solve for \( x \) to find points to plot. For example, if \( y = 1 \), then \( x = -\frac{1}{3} \), giving the point (\(-\frac{1}{3}, 1\)) on the graph.
05
Verify Graph Symmetry
Check that the graph is symmetric across the x-axis because the equation involves \( y^2 \). For every point \((x, y)\), the corresponding point \((x, -y)\) should also be on the graph.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Parabola Orientation
Understanding the orientation of a parabola is crucial when graphing these curves. A parabola can open upwards, downwards, left, or right. The direction depends on the equation's terms. For the given equation, \( y^2 = -\frac{1}{3}x \), it's important to notice the squared term: \( y^2 \). This leads to a horizontal parabola.
- When \( y^2 = kx \), it opens horizontally.
- If \( k > 0 \), the parabola opens to the right.
- If \( k < 0 \), as in our equation, it opens to the left.
Vertex of the Parabola
The vertex is a key point on a parabola and acts as its "peak" or "valley," depending on orientation. For our equation \( y^2 = -\frac{1}{3}x \), this special point is the origin (0,0).
- A parabola in the form \( y^2 = kx \) or \( x^2 = ky \) without additional terms is centered at the origin.
- If a parabola has added constants, such as \( y^2 = -\frac{1}{3}(x - h) + k \), the vertex would shift to \((h, k)\).
Equation Symmetry
Symmetry in a parabolic graph ensures that for any point on the parabola, there is a directly corresponding point equidistant from the axis of symmetry. Our equation \( y^2 = -\frac{1}{3}x \) indicates symmetry around the x-axis.
- The equation contains \( y^2 \), suggesting reflection across the x-axis.
- For example, if a point \((x, y)\) exists on the graph, then \((x, -y)\) must also be on it.
Graphing Techniques
Graphing a parabola involves several key steps to ensure accuracy. Leveraging these techniques can aid in achieving a precise sketch.
- First, identify and plot the vertex, such as (0, 0) in our case.
- Next, determine the orientation and open the parabola accordingly. Our example opens to the left due to the equation's negative sign with \( x \).
- Select values of \( y \), substitute them into the equation, and calculate corresponding \( x \)-values. For example, if \( y = 1 \), then \( x = -\frac{1}{3} \), giving the point \((-\frac{1}{3}, 1)\).
- Verify symmetry by checking opposite points, like \((-\frac{1}{3}, -1)\) for \((-\frac{1}{3}, 1)\).
