Problem 47 - Algebra and Trigonometry

Question

(a) Find equations for the family of parabolas with vertex at the origin and with directrixes \(y=\frac{1}{2}, y=1, y=4\) and \(y=8\). (b) Draw the graphs. What do you conclude?

Step-by-Step Solution

Answer

The equations are \(x^2 = y, x^2 = 2y, x^2 = 8y, x^2 = 16y\). Parabolas become narrower as the directrix moves upwards.

Step 1: Understanding the Parabola Equation

A parabola's equation in vertex form is typically represented as \( (x - h)^2 = 4p(y - k) \) where \((h, k)\) represents the vertex. Here, the vertex is at the origin \((0, 0)\), so the equation simplifies to \( x^2 = 4py \). The parameter \(p\) is the distance from the vertex to the focus and also to the directrix. For each directrix, \(y = k\), \(p\) will be \(-k/2\) since the directrix is above the vertex.

Step 2: Derive Equations for Each Directrix

For each directrix given, calculate \(p\) and substitute it into the parabola equation:- **Directrix**: \(y = \frac{1}{2}\) - Here, \(p = -\frac{1}{4}\), so the equation is \(x^2 = y\).- **Directrix**: \(y = 1\) - Here, \(p = -\frac{1}{2}\), so the equation is \(x^2 = 2y\).- **Directrix**: \(y = 4\) - Here, \(p = -2\), so the equation is \(x^2 = 8y\).- **Directrix**: \(y = 8\) - Here, \(p = -4\), so the equation is \(x^2 = 16y\).

Step 3: Graph Each Parabola

Plot each parabola on the same set of axes:- The first parabola with directrix \(y = \frac{1}{2}\) will appear as the shallowest one.- As the value of \(y\) increases for the directrix, the parabolas become wider and more vertically stretched (narrower). The second parabola \(x^2 = 2y\) is narrower than the first.- This pattern continues with the third \(x^2 = 8y\) and fourth \(x^2 = 16y\) parabola becoming progressively narrower.

Step 4: Analyze the Graphs

On drawing the graphs, you will notice that all parabolas open upward, sharing the same vertex at the origin. The greater the value of the directrix's \(y\)-position, the more narrow the parabola becomes, indicating that the distance \(p\) governs the width of the parabola.

Key Concepts

Vertex Form

The vertex form of a parabola is a convenient way to express the equation of a parabola, particularly when you know the vertex's coordinates. A parabola is a U-shaped curve that can open up, down, left, or right depending on its equation. The standard vertex form equation is

• \((x - h)^2 = 4p(y - k)\) for vertical parabolas
• \((y - k)^2 = 4p(x - h)\) for horizontal parabolas
The parameters
• \((h, k)\) represent the vertex, the point where the parabola changes direction.
• \(p\) determines the distance from the vertex to the focus, influencing the parabola's width.
For this exercise, since the vertex is at the origin \((0, 0)\), the equation simplifies to
• \(x^2 = 4py\), making calculations easier.
Understanding the vertex form helps you manipulate the equation to fit specific conditions, such as a given vertex or directrix.

Directrix

The directrix is a fundamental feature of a parabola. It is a fixed line used to define and construct the parabola along with the focus.

• The role of the directrix is to help maintain a constant distance to any point on the parabola compared to the focus.
• For upright parabolas, it is a horizontal line. For sideways parabolas, it is vertical.

Using the directrix, you can deduce the parameter \(p\), which is vital for forming the parabola's equation.

• In this exercise, given directrix lines like \(y = \frac{1}{2}\), the calculation involves finding \(p\) as \(-\frac{1}{4}\), showing that the directrix lies above the vertex while the parabola opens upwards.

This relationship ensures that each point on the parabola has an equidistant line from both the directrix and the corresponding focus.

Distance to Focus

The distance to the focus from the vertex (denoted by \(p\)) is crucial in determining the shape of a parabola.

• The value of \(p\) implies how far the focus is from the vertex along the parabola's axis of symmetry.
• It assists in determining the parabolic path by indicating how much the curve opens.

For the problem at hand:

• A small \(p\) results in a wider or "shallow" parabola, seen in the equation \(x^2 = y\) where \(p = -\frac{1}{4}\).
• A larger negative \(p\) means a narrow parabola, such as \(x^2 = 16y\) where \(p = -4\).

In essence, the value of \(p\) offers insight into the "spread" of the parabola by dictating both its wideness and closeness to the focus.

Graphing Parabolas

Graphing parabolas can visually demonstrate how changes in their equation affect their shape and orientation.

• Each equation derived from a specific directrix allows for plotting the corresponding parabola.
• In this exercise, plotting shows that as the directrix value moves higher (greater \(y\) values), the parabola becomes narrower.

When graphing:

• Always start by plotting the vertex point.
• Draw the axis of symmetry, which is vertical when \(x^2 = 4py\).
• Plot key points on either side of the vertex to see how quickly the parabola opens or stretches.
• Compare how the different values of \(p\) affect the resulting curve shapes.

Graphing these parabolas reinforces the relationship between the algebraic equation and the graphical representation.