Problem 65 - 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 and shifting \(p\) causes the parabola to widen or close with changes in vertex proximity.

Step 1: Investigate the Parabola Equation

The equation given is \(x^2 = 4p(y+p)\). This fits the standard form \(x^2 = 4a(y-k)\), indicating a vertical parabola that opens upwards if \(a > 0\) and downwards if \(a < 0\). The goal is to verify these characteristics for different \(p\) values.

Step 2: Identify the Vertex and Focus

For the standard form \(x^2 = 4a(y-k)\), the vertex is at \((0, -p)\) and the focus is at \((0, 0)\), as derived from the equation when rewritten as \(x^2 = 4p(y+p-p)\), confirming that the focus is at the origin.

Step 3: Graph the Parabolas

Plot the parabolas for each \(p\) value given: \(-2, -\frac{3}{2}, -1, -\frac{1}{2}, \frac{1}{2}, 1, \frac{3}{2}, 2\). Each parabola will have its vertex at \((0, -p)\). Notice as \(p\) changes, the perceived width and direction (upward or downward) of the parabola will change.

Step 4: Analyze Vertex Proximity to the Origin

As the vertex \((0, -p)\) moves closer to the origin, the distance \(|p|\) decreases. This results in the parabola becoming less "steep" and shallower in appearance if the magnitude of \(p\) decreases. When \(p=0\), the parabola is at its shallowest and symmetric about the axis of symmetry running through the origin.

Step 5: Describe the Effect

As \(p\) approaches zero, the parabolas become wider and less deep. The vertex moving closer to the origin suggests the parabola becomes broader. The change in \(p\) affects how open or closed the parabola appears. The direction (up/down) depends on the sign of \(p\).

Key Concepts

Focus of a Parabola

The focus of a parabola is a key element that helps define its shape along with the directrix. In the confocal parabolas described in the exercise, all the parabolas share a common focus, which is positioned at the origin, \(0, 0\). The focus is one of the defining characteristics of a parabola, determining how "open" the shape is with respect to the central axis.
In mathematical terms, for the specific given equation \(x^2 = 4p(y+p)\), the focus remains constant at \(0, 0\). This means all parabolas in this system are confocal, with their focal point remaining unchanged regardless of the varying value of \(p\). A stable focus suggests that the maximum distance from the parabola to the focus directly influences the steepness and width of the parabolas in the family."

Vertex of a Parabola

The vertex of a parabola is the point where the curve reaches its maximum or minimum value, depending on its orientation. In the given parabolic equation \(x^2 = 4p(y+p)\), the vertex can be identified at the point \(0, -p\).

• When \(p\) is positive, the parabola opens downwards and the vertex sits below the origin.
• When \(p\) is negative, it opens upwards, moving the vertex above the origin.

The vertex is crucial in understanding the parabola's orientation. For this exercise, notice as \(p\) gets smaller in absolute value, the vertex moves closer to the origin, making the curve flatter and less pronounced. Understanding the role of the vertex aids in anticipating how parabolas behave graphically when their defining parameters, like \(p\), change.

Graphing Parabolas

When graphing parabolas, identifying the respective vertex and focus is essential. Graphing the given family of parabolas \(x^2 = 4p(y+p)\) across varying \(p\) values demonstrates how changes in \(p\) translate to changes in the graph. You need to map out each vertex at \(0, -p\) and draw the opening accordingly:

• If \(p > 0\), the parabola opens downwards.
• If \(p < 0\), it opens upwards.

This consistent focus point leads to unique yet predictable parabola formations on the graph. As \(p\) becomes smaller in magnitude, the parabolas appear wider and less steep. Practicing with these parabolic transformations strengthens spatial understanding of how algebraic changes affect geometric representations.

Parabola Equation

The equation \(x^2 = 4p(y+p)\) represents a vertical parabola, a fundamental shape in algebra that can open either upwards or downwards based on the value of \(p\). The equation structure itself provides guidelines to determine key features of the parabola without graphing:

• The term \(4p\) dictates the width of the parabola. When \(4p\) increases, the parabola becomes narrower.
• The sign of \(p\) identifies the direction the parabola opens (up if negative, down if positive).

This equation aligns with the vertex form \(x^2 = 4a(y-k)\) in standard notation, making it easy to calculate both the vertex at \(0, -p\) and focus. Understanding these components empowers you to construct accurate graphs and interpret the geometric layout efficiently.