Problem 14 - Precalculus: Mathematical for Calculus

Question

Find the focus, directrix, and focal diameter of the parabola, and sketch its graph. $$x=\frac{1}{2} y^{2}$$

Step-by-Step Solution

Answer

Focus: \((\frac{1}{2}, 0)\), Directrix: \(x = -\frac{1}{2}\), Focal Diameter: 2.

Step 1: Identify the Parabola Form

The given equation is \( x = \frac{1}{2} y^2 \). This equation represents a parabola that opens horizontally, specifically to the right if the coefficient of \( y^2 \) is positive. The standard form for a horizontal parabola is \( (y-k)^2 = 4p(x-h) \). Thus, \( k = 0\), \( h = 0\), and \( 4p = 2 \).

Step 2: Solve for p

From the standard form \((y-k)^2 = 4p(x-h)\), we identified that \(4p = 2\). Thus, \(p = \frac{1}{2}\). This value of \(p\) will help us find the focus and directrix.

Step 3: Determine the Vertex

The vertex \((h, k)\) of the parabola is \((0, 0)\) since the equation is in the form \((y - k)^2 = 4p(x-h)\) and both \(h\) and \(k\) are 0.

Step 4: Find the Focus

For a horizontally opening parabola, the focus is \((h + p, k)\). Substituting the values, we get \( (0 + \frac{1}{2}, 0) = (\frac{1}{2}, 0)\).

Step 5: Determine the Directrix

The directrix of a horizontally opening parabola is given by the line \( x = h - p \). Using our \(h\) and \(p\), we find the directrix as \( x = 0 - \frac{1}{2} = -\frac{1}{2} \).

Step 6: Calculate the Focal Diameter

The focal diameter is the length of the line segment through the focus perpendicular to the axis of symmetry of the parabola. For the parabola \((y-k)^2 = 4p(x-h)\), the focal diameter is \(|4p|\), which is \(|2|=2\).

Step 7: Sketch the Parabola

To sketch the parabola, plot the vertex at \((0, 0)\), focus at \((\frac{1}{2}, 0)\), and the directrix line at \(x = -\frac{1}{2}\). The parabola opens to the right, having a relatively wider spread because the coefficient of \(y^2\) is small. Draw it as a symmetric curve centered on the x-axis.

Key Concepts

Horizontal Parabola

A horizontal parabola is a type of parabola that opens to the left or right rather than upwards or downwards. In the equation \( x = \frac{1}{2} y^2 \), we see a relation where the variable \( x \) depends on \( y^2 \). This signifies the parabola opens horizontally. Specifically, it opens to the right because the coefficient of \( y^2 \) is positive. Horizontal parabolas can be represented in the standard form \( (y-k)^2 = 4p(x-h) \). Here, \( (h, k) \) is the vertex of the parabola, and \( p \) determines the distance from the vertex to the focus or the directrix. A positive \( p \) indicates an opening to the right, and a negative \( p \) would indicate an opening to the left. Understanding the orientation of the parabola is crucial for graphing and determining properties like the focus and directrix of the parabola.

Vertex Form of Parabola

The vertex form of a parabola simplifies identifying key features, especially the vertex, from the equation. For horizontal parabolas, this form is \( (y - k)^2 = 4p(x - h) \). In this form:

• \( h \) and \( k \) represent the x and y coordinates of the vertex, respectively.
• \( p \) dictates how far and in which direction the parabola shifts from the vertex towards the focus and directrix.

In our given equation \( x = \frac{1}{2} y^2 \), comparing it with \( (y-k)^2 = 4p(x-h) \), we see that \( h = 0 \), \( k = 0 \), making the vertex \((0, 0)\). The term \( 4p = 2 \) gives us \( p = \frac{1}{2} \). This simple transformation helps in easily plotting and understanding the behavior of the parabola on a graph.

Focal Diameter

The focal diameter, also known as the latus rectum, is an essential feature of a parabola. It is the length of the line segment perpendicular to the axis of symmetry that passes through the focus. The focal diameter provides insights into the width and shape of the parabola. For any parabola in the form \( (y-k)^2 = 4p(x-h) \), the focal diameter is \(|4p|\). In our case, from the equation \( x = \frac{1}{2} y^2 \), we identified \( 4p = 2 \), implying the focal diameter is \( |2| = 2 \). This measure tells us how far apart the sides of the parabola are as they curve away from the vertex. A larger focal diameter indicates a wider curve, whereas a smaller one suggests a sharper curve. The focal diameter is a crucial part of understanding the geometry of the parabola in space and aids in graphing it more accurately.