Problem 21 - Precalculus: Mathematical for Calculus

Question

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

Step-by-Step Solution

Answer

The parabola opens to the left with vertex at (0,0).

Step 1: Identify the Equation Type

The given equation is in the form \[y^2 = -\frac{1}{3}x\]. This is the equation of a parabola with its axis parallel to the x-axis.

Step 2: Determine the Orientation

Since the equation is \[y^2 = -\frac{1}{3}x\], the parabola opens to the left because the coefficient of \(x\) is negative.

Step 3: Find the Vertex

The standard form \[4p(y-k)^2 = (x-h)\] has its vertex at \((h, k)\).For \[y^2 = -\frac{1}{3}x\], the vertex is at \((0, 0)\).

Step 4: Graph the Parabola

To graph the parabola, note that the vertex is at (0,0) and it opens to the left.The distance \(p\) from the vertex to the focus (also controlling the width) is found from comparing to \[4p(y-k)^2 = (x-h)\].Here, \(4p = -\frac{1}{3}\), giving \(p = -\frac{1}{12}\). Plot the vertex and sketch a curve opening leftwards, becoming wider as it moves away from the origin.

Key Concepts

Graphing Parabolas

Graphing parabolas can initially seem challenging, but with a bit of practice, it becomes fairly straightforward. It's important to recognize the standard form of a parabolic equation. Parabolas that have the form \(y^2 = ax\) or \(x^2 = ay\) have their axes parallel to either the x-axis or y-axis respectively. In our case, the equation \(y^2 = -\frac{1}{3}x\) suggests the axis is parallel to the x-axis.

To begin graphing this parabola, you should first find the vertex, often a starting point in plotting curves. Then determine the orientation for how the parabola opens based on the equation. With these initial points understood, plotting becomes easier. You can then scale and sketch accordingly, considering factors like the focal distance that will alter the width of the curve.

Orientation of Parabolas

The orientation of a parabola is determined by the sign and position of the coefficients in its equation. Specifically, in the equation \(y^2 = -\frac{1}{3}x\), note how \(x\) is isolated on one side, meaning the parabola's axis runs parallel to the x-axis.

• If the coefficient of \(x\) is negative, as in our case, the parabola opens to the left.
• If it were positive, it would open to the right.

Considering the given equation, we note that the parabola opens to the left. This knowledge helps significantly when plotting the curve initially by guiding how to position the parabola on the coordinate plane. From here, knowing more about the vertex and focus allows an accurate depiction.

Vertex of Parabola

The vertex of a parabola is a crucial point, often marking the peak for those opening upwards or downwards, or the farthest point left or right for those horizontally oriented. For the equation \(y^2 = -\frac{1}{3}x\), the standard form becomes clear: \((y-k)^2 = -\frac{1}{3}(x-h)\).

Since \(h\) and \(k\) are zero in this example, the vertex is at the origin \((0, 0)\).

• The vertex serves as a pivot point from which the parabolic curve expands.
• This point also plays a role in symmetry; parabolas are always symmetric with respect to their axis running through the vertex.

Understanding the location of the vertex aids in correctly plotting and balancing the parabola on the graph.

Focus of Parabola

The focus of a parabola is another key point that complements the vertex. It's important for understanding the parabola’s shape and how it directs points. In any parabola, all points are equidistant from the focus and the directrix. This equidistance property is fundamental to the parabola's definition.

For the equation \(y^2 = -\frac{1}{3}x\), to find the focus, we use the relation \(4p = -\frac{1}{3}\), giving \(p = -\frac{1}{12}\). It's positioned along the axis of symmetry by a distance of \(p\) from the vertex.

• Since \(p\) is negative, and the parabola opens left, the focus is at \((-\frac{1}{12}, 0)\).
• This proximity to the vertex means that the arms of the parabola are quite wide and not sharply curved.

Graphing involves plotting this focus point along with the vertex, guiding you in curating the precise path as the parabola extends.