Problem 52 - Precalculus Mathematics for Calculus

Question

(a) Find equations for the family of parabolas with vertex at the origin, focus on the positive \(y\) -axis, and with focal diameters \(1,2,4,\) and 8 (b) Draw the graphs. What do you conclude?

Step-by-Step Solution

Answer

Equations: \( y = x^2 \), \( y = \frac{1}{2}x^2 \), \( y = \frac{1}{4}x^2 \), \( y = \frac{1}{8}x^2 \). Wider parabolas correspond to larger focal diameters.

Step 1: Formula for the Focus and Directrix of a Parabola

The standard form of a parabola with vertex at the origin and focus on the positive y-axis is \( y = \frac{1}{4p}x^2 \), where \( p \) is the distance from the vertex to the focus. The focal diameter of the parabola, which is the length of the latus rectum, is equal to \( 4p \). We use this information to find equations for different values of focal diameter.

Step 2: Find Equations for Each Focal Diameter

Given the focal diameters of 1, 2, 4, and 8, we need to find \( p \) for each. For a focal diameter of 1, \(4p = 1\) gives \( p = \frac{1}{4} \). The equation becomes \( y = x^2 \). Similarly, for a focal diameter of 2, \( p = \frac{1}{2} \): \( y = \frac{1}{2}x^2 \), for a focal diameter of 4, \( p = 1 \): \( y = \frac{1}{4}x^2 \), and for a focal diameter of 8, \( p = 2 \): \( y = \frac{1}{8}x^2 \).

Step 3: Draw the Graphs

Draw each parabola based on the equations derived. For each parabola, plot points that satisfy the equation, ensuring they open upwards from the vertex. You will notice that as the focal diameter increases, the parabola becomes wider.

Step 4: Conclusion on the Graphs' Behavior

The graphs show that as the focal diameter (or \( p \)) increases, the parabolas become wider. Thus, the focal diameter influences the spread of the parabola; larger diameters result in wider opening parabolas.

Key Concepts

Vertex

The vertex is a critical point on a parabola. It represents the turning point where the curve changes direction. In a parabola, the vertex can be thought of as the 'starting point.' For the exercise given, the vertex is located at the origin, \((0,0)\).An easy way to visualize this is to think of the vertex as the tip of the U-shaped curve.

• The vertex is either the highest or lowest point on the graph, depending on the parabola's orientation.
• It serves as the midpoint between the focus and the directrix, although in our exercise the directrix is not explicitly mentioned.

Understanding the vertex helps you easily plot the parabola on a graph, as it dictates the initial placement before considering other elements like the focus and latus rectum.Continuing from the vertex, the parabola curves either upwards or downwards, depending on its equation.

Focus

The focus of a parabola is one of its defining properties. It is a specific point located inside the curve, which, along with the directrix, helps to 'direct' the shape of the parabola. In the case of a parabola that opens upwards, like in our exercise, the focus is positioned vertically above the vertex along the y-axis.
The distance between the vertex and the focus is known as \(p\), and for our exercise, the parabolas have varying \(p\) values depending on their focal diameters.

• For a larger \(p\) value, the focus is further from the vertex, causing the parabola to open wider.
• The closer the focus is to the vertex, the tighter the parabola curves.

By determining the position of the focus, students can gain insight into how the shape of the parabola will spread out.

Focal Diameter

The focal diameter, also known as the focal width or latus rectum, is the length of the line segment perpendicular to the axis of symmetry of the parabola that passes through the focus. It reflects how 'open' or 'wide' the parabola is.
In the formula \(4p\), the focal diameter is equal to four times the distance from the vertex to the focus. This exercise gives us values for the focal diameter, such as 1, 2, 4, and 8.

• A focal diameter of 1 corresponds to a parabola that is tightly curved.
• As the focal diameter increases to 8, the parabola becomes wide open.

Analyzing the focal diameter helps students understand the expansiveness of the parabola. By calculating \(p\) from the given focal diameter, students can derive the equation for each parabola and predict its shape and openness before graphing it.

Latus Rectum

The latus rectum is a line segment that is critical in defining the shape and orientation of a parabola. It is directly tied to the focus and runs perpendicular to the axis of symmetry. In this context, it helps in visualizing the openness of the parabola.
Remember, the length of the latus rectum equals the focal diameter, which has been denoted as \(4p\) in mathematical terms. This means that the properties of the latus rectum will vary based on \(p\).

• The latus rectum helps confirm that a parabola is symmetrical around its vertex.
• It serves to define how much a parabola 'spreads' horizontally.

With different \(p\) values in this exercise, the latus rectum changes size, directly affecting the width of the parabola.By examining the latus rectum, students can get a clearer picture of how the dimensions and shape of the parabola change with varying parameters.