Problem 62 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the vertex, focus, and directrix of the parabola and sketch its graph. Use a graphing utility to verify your graph. $$y=-2 x^{2}$$

Step-by-Step Solution

Answer

The vertex of the parabola is at (0,0), the focus is at (0,-1/8), and the equation of the directrix is \(y=1/8\). The parabola opens downwards with a narrow curve due to the value of a being -2.

Step 1: Find the Vertex

The vertex form of a parabola is \(y=a(x-h)^2 + k\), where \((h, k)\) is the vertex. In the given equation \(y=-2x^2\), h = 0 and k = 0 because it is already simplified. Therefore, the vertex of the equation is at the point (0,0).

Step 2: Find the Focus and Directrix

The focus of the parabola can be determined using the formula \(a = 1/(4f)\), where a is the coefficient of \(x^2\) and f is the focal length. In the equation \(y = -2x^2\), a is equal to -2. Solving the equation for f gives \(f = 1/(4*-2) = -1/8\), which represents the distance from the vertex to the focus along the axis of symmetry. The focus is located below the vertex because a is negative, so the focus of the parabola is (0, -1/8).\nThe directrix of a parabola is a horizontal line that is |1/4a| units away from the vertex and lies on the side opposite the focus. Since a is negative, the directrix will be located above the vertex. Therefore, the equation of the directrix is \(y = 1/8\).

Step 3: Sketch the Graph

Plot the vertex at the origin (0,0). After that, plot the focus at point (0, -1/8). Then, draw a horizontal line at \(y=1/8\) to represent the directrix. The parabola will open downwards because a is negative. Draw the parabola such that it is narrow due to the value of a being -2 and extending infinitely in both the positive and negative x directions and verify your graph using a graphing utility.

Key Concepts

Vertex

The vertex of a parabola is a significant point that defines the minimum or maximum point of the curve. This is where the parabola changes direction. For the equation of a parabola in vertex form, which looks like \[ y = a(x-h)^2 + k \]the vertex is located at the point \((h, k)\). In our given equation \(y = -2x^2\), it is already in a simplified form where \(h = 0\) and \(k = 0\).
Hence, the vertex of the parabola is at the origin, specifically at the point \((0, 0)\). In a graph, this would be where the parabola peaks or dips, depending on the orientation of the curve. When the parabola opens downward, like in this case, the vertex represents the highest point of the graph.

Focus

The focus of a parabola is a crucial point that defines how "bent" the parabola is. Every point on the parabola is equidistant from the focus and the directrix. To find the focus for the parabola given by \(y = -2x^2\), we need to calculate the focal length \((f)\) using the formula:\[ a = \frac{1}{4f} \]
Here, \(a = -2\), so:\[ f = \frac{1}{4(-2)} = -\frac{1}{8} \]
This negative focal length indicates the focus is beneath the vertex along the y-axis. Therefore, for this parabola, the focus is located at the point \((0, -\frac{1}{8})\). This suggests that the parabola's curve bends such that it seems to "focus" on this point, which is positioned directly beneath the vertex.

Directrix

The directrix is a guideline in understanding the definition of a parabola. It's a line that runs parallel to the axis of symmetry. The property of a parabola is such that each point on the parabola maintains an equal distance from both the focus and the directrix.
Given the parabola's equation \(y = -2x^2\), the directrix is a horizontal line above the vertex, since \(a\) is negative and the parabola opens downward. The formula for the directrix is:\[ y = k + \frac{1}{4|a|} \]Since the vertex \((h, k)\) is \((0, 0)\), the directrix lies at:\[ y = 0 + \frac{1}{4|2|} = \frac{1}{8} \]
Thus, the directrix can be expressed as the line \(y = \frac{1}{8}\). On a graph, it would appear as a horizontal line placed at an equal distance from the vertex but on the opposite side from the focus.

Axis of Symmetry

The axis of symmetry for a parabola is an imaginary vertical line that divides the parabola into two mirrored halves. Essentially, it's the line that runs right through the vertex, determining the reflective property of a parabola's shape.
For the parabola \(y = -2x^2\), since it's symmetric about the y-axis, the axis of symmetry is the line:\[ x = h \]In this case, \(h = 0\), so the axis of symmetry is:\[ x = 0 \]
This axis is very useful in graphing and analyzing the behavior of the parabola, as it assists in predicting the shape and direction of the curve. Observing this symmetry line, you will see that every point on one side of the parabola has a mirror point on the opposite side, denoting the uniform spread of the parabola about this axis.