Problem 41 - Precalculus Mathematics for Calculus

Question

This exercise deals with confocal parabolas, that is, families of parabolas that have the same focus. (a) Draw graphs of the family of parabolas $$x^{2}=4 p(y+p)$$ for \(p=-2,-\frac{3}{2},-1,-\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, 2\) (b) Show that each parabola in this family has its focus at the origin. (c) Describe the effect on the graph of moving the vertex closer to the origin.

Step-by-Step Solution

Answer

All parabolas have their focus at the origin, regardless of \( p \). Varying \( p \) moves the vertex and changes the parabola's width.

Step 1: Rewrite the Parabola Equation

The given equation is \( x^2 = 4p(y+p) \). To find the vertex, rewrite it in the standard form \( (x-h)^2 = 4p(y-k) \), where \( (h,k) \) is the vertex. For \( x^2 = 4p(y+p) \), this simplifies to \( (x-0)^2 = 4p(y+p) \) and hence the vertex is \((0, -p)\).

Step 2: Identify the Focus for Each p

For any parabola of the form \( (x-h)^2 = 4p(y-k) \), the focus is at \((h, k+p)\). Since our equation is \( x^2 = 4p(y+p) \), the focus is at \((0, -p+p) = (0, 0)\). Thus, each parabola has its focus at the origin, regardless of the value of \( p \).

Step 3: Effect of Varying p on Graphs

As you change the value of \( p \), you affect the vertex of the parabola, which is at \((0, -p)\). Lowering the absolute value of \( p \) (moving \( p \) towards zero) brings the vertex closer to the origin. This causes the parabola to become narrower as \( |p| \) decreases and wider as \( |p| \) increases, since \( 4p \) is the focal width.

Step 4 (Optional): Graph Visualization

For each \( p \) value (\( -2, -\frac{3}{2}, -1, -\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, 2 \)), sketch the corresponding parabolas. Notice how the shape of the parabola changes from wide to narrow as \( |p| \) decreases and verify that all parabolas have their focus at the origin.

Key Concepts

Confocal Parabolas

Confocal parabolas are a fascinating family of curves in geometry where all members share the same focus. This unique characteristic sets confocal parabolas apart from other parabolic families. Let's picture a scenario: you have multiple parabolas, all with their tips pointing upwards or downwards, and despite how they might appear in different shapes or sizes, they converge at one point—their focus. In our exercise, we analyze parabolas given by the equation:\[ x^2 = 4p(y+p) \]Here, the parameter \( p \) changes, yet every parabola in this family focuses at the origin \((0,0)\). These parabolas not only share a common focus but also have their vertex positions influenced by the parameter \( p \). Exploring confocal parabolas helps us understand how a shared geometric feature can unify multiple shapes.

Parabola Geometry

Parabolas are open curves forming part of the conic sections family. They have well-defined geometric properties that make them unique.

• **Vertex:** The vertex is the turning point of the parabola. It's the point where the curve changes from opening downward to upward or vice versa. For the parabolas given by our equation, the vertex is located at \((0, -p)\).
• **Axis of Symmetry:** This vertical line passes through the vertex and divides the parabola into two mirror-image halves. In our case, it's the y-axis \(x=0\).
• **Focal Width:** This is a measure of how "wide" a parabola is, determined by \(4p\). As the magnitude of \(p\) changes, so does the tightness or openness of the parabola's curve.

In essence, the geometry of a parabola is characterized by its vertex position, axis of symmetry, and focal width. All these elements work together to describe the shape and orientation of the parabola.

Focus and Vertex of Parabola

Understanding the focus and vertex of a parabola is crucial to analyzing its shape thoroughly. The vertex is the point closest to the focus, which for the given family of parabolas represented as:\[ x^2 = 4p(y+p) \]lies at \((0, -p)\). No matter the value of \(p\), the focus for all these parabolas is always centered at the origin \((0,0)\). This central focus point is significant because it is where distances to any point on the parabola are consistently shortest along the directrix-axis-focus framework.

• **Focus:** This is a fixed point used to define and construct the parabola. For our family of parabolas, it remains steadfast at \((0,0)\) regardless of changes in \(p\). This stability in their focus location is what makes them confocal parabolas.
• **Vertex Movement:** By adjusting \(p\), the vertex shifts up and down along the y-axis, positioned at \((0, -p)\). As the vertex approaches the origin, the parabola's openness modifies: reducing \(p\) values tighten the curve, while increasing \(|p|\) results in a broader shape.

Grasping the relationship between the focus and vertex helps decode the behavior and configuration of parabolas, making it easier to visualize and graph these conic sections effectively.