Problem 6 - Precalculus with Calculus Previews

Question

Find the vertex, focus, directrix, and axis of the given parabola. Graph the parabola. \(x^{2}=\frac{1}{10} y\)

Step-by-Step Solution

Answer

Vertex: (0,0); Focus: (0,1/40); Directrix: y=-1/40; Axis: x=0.

Step 1: Identify the Standard Form

The given parabola equation is \(x^2 = \frac{1}{10}y\). This is similar to the standard form \(x^2 = 4py\). Our goal is to identify \(p\).

Step 2: Find the Value of p

By comparing \(x^2 = 4py\) to \(x^2 = \frac{1}{10}y\), we see that \(4p = \frac{1}{10}\). Solving for \(p\), we find \(p = \frac{1}{40}\).

Step 3: Determine the Vertex

In the standard form \(x^2 = 4py\), the vertex of the parabola is at the origin \((0, 0)\) because there are no extra terms added to \(x\) or \(y\).

Step 4: Calculate the Focus

The focus of the parabola is at \((0, p)\). Substituting \(p = \frac{1}{40}\), the focus is at \(\left(0, \frac{1}{40}\right)\).

Step 5: Determine the Directrix

The directrix of the parabola \(x^2 = 4py\) is the line \(y = -p\). Therefore, the directrix is \(y = -\frac{1}{40}\).

Step 6: Identify the Axis of Symmetry

The axis of symmetry for a parabola with the equation \(x^2 = 4py\) is the \(y\)-axis, which is the vertical line \(x = 0\).

Step 7: Graph the Parabola

To sketch the graph, plot the vertex at \((0,0)\), the focus at \((0, \frac{1}{40})\), and draw the directrix as a horizontal line at \(y = -\frac{1}{40}\). The parabola opens upwards, and it is symmetric about the \(y\)-axis.

Key Concepts

Vertex

The vertex is a crucial concept when studying parabolas. It is the point where the parabola changes direction and is often considered the "starting point" of the parabola's curve. For the equation given, which is

• \(x^2 = \frac{1}{10}y\),
• we can observe that the parabola is already in a form close to the standard parabolic form \(x^2 = 4py\).

When expressed in this format without additional terms on \(x\) or \(y\), the vertex is located at the origin. Therefore, the vertex for this parabola is

• at the point \((0, 0)\).

This location is essential as it provides a reference point to find other related features of the parabola.

Focus

The focus of a parabola is another vital element that defines its shape. The focus lies inside the parabola and is used to guide its "bowl-like" structure. For the parabola given by

• \(x^2 = \frac{1}{10}y\),
• we find its equivalent standard form as \(x^2 = 4py\), where \(4p = \frac{1}{10}\).

Solving for \(p\), we determine that \(p = \frac{1}{40}\). In this configuration, the focus of the parabola is situated along the axis of symmetry, which we will discuss later, at the point

• \((0, \frac{1}{40})\).

The presence of the focus helps in demonstrating the parabola's depth and how steeply it opens.

Directrix

The directrix of a parabola provides an essential geometry that works with the focus to define the shape of the parabola. It is a line that is perpendicular to the axis of symmetry and situated opposite the focus. In this parabolic equation

• \(x^2 = \frac{1}{10}y\),
• we have determined \(p = \frac{1}{40}\).

This result lets us identify the directrix. Since the focus is at \(y = \frac{1}{40}\), the directrix can be found at

• y = -\frac{1}{40},

which runs horizontally. Understanding the directrix helps you grasp the full width and opening directions of the parabola and how it is symmetrically balanced with the focus.

Axis of Symmetry

The axis of symmetry for a parabola is the vertical line that cuts it into two mirror-image halves. It is significant because it passes through both the vertex and the focus. For the parabola denoted by

• \(x^2 = \frac{1}{10}y\),
• its shape is vertically aligned, meaning the axis of symmetry is the y-axis.

Hence, the axis of symmetry is the line

• x = 0,

which shows that the parabola is symmetric around the y-axis. Recognizing this line of symmetry is critical for predicting the shape and direction of the parabola's arms, ensuring a clear understanding of its geometric properties.