Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Give the focus, directrix, and axis of each parabola. $$x^{2}=4 y$$

Step-by-Step Solution

Answer

Focus: (0, 1); Directrix: y = -1; Axis of symmetry: x = 0.

Step 1: Identify the Parabola Form

The given equation is in the form \(x^2 = 4py\), which indicates that the parabola is vertical and opens either up or down. Here, the coefficient of \(y\) is 4, meaning \(4p = 4\).

Step 2: Solve for Parameter p

From the equation \(4p = 4\), divide both sides by 4 to solve for \(p\). This gives us \(p = 1\).

Step 3: Determine the Focus

For a parabola expressed as \(x^2 = 4py\), the focus is given by the point \((0, p)\). Since \(p = 1\), the focus is \((0, 1)\).

Step 4: Determine the Directrix

The directrix of a parabola \(x^2 = 4py\) is a horizontal line given by \(y = -p\). With \(p = 1\), the directrix is the line \(y = -1\).

Step 5: Identify the Axis of Symmetry

For a vertically oriented parabola \(x^2 = 4py\), the axis of symmetry is the vertical line that passes through the vertex. It is the line \(x = 0\), also the y-axis.

Key Concepts

Focus of a Parabola

The focus of a parabola is a pivotal concept in understanding its geometric properties. In basic terms, the focus is a specific fixed point inside the parabola. This point and the directrix work together to define the parabola's shape.
For a parabola in the form of \(x^2 = 4py\), the focus is located at \((0, p)\). In this context, \(p\) represents the distance from the vertex to the focus along the axis of symmetry.
In our exercise, with the equation \( x^2 = 4y \), value of \(p\) is 1. Hence, the focus is at \((0, 1)\).
This focus point is essential because it is where all lines perpendicular to the directrix and parallel to the axis of symmetry seem to converge. Understanding where the focus lies helps in drawing the precise shape of the parabola.

Directrix of a Parabola

The directrix of a parabola serves as a guideline, helping to define its structure. This is a line that lies outside the parabola, and every point on the parabola is equidistant from both this line and the focus.
In our parabola's form, \(x^2 = 4py\), the directrix can be represented by a horizontal line \(y = -p\). This line is parallel to the \(x\)-axis and beneath the vertex of the parabola.

• For the equation \(x^2 = 4y\), where \(p = 1\), the directrix is the line \(y = -1\).

The relationship between the focus and the directrix is crucial in the reflective property of the parabola. Any point on the parabola is equidistant from both the focus and this directrix line, forming the core geometric definition of the parabola.

Axis of Symmetry

The axis of symmetry in a parabola is a critical concept because it visually divides the parabola into two mirror images of each other.
Let's consider a parabola with the equation \(x^2 = 4py\). Here, the axis of symmetry is a vertical line that passes through the vertex, and it is always parallel to the \(y\)-axis. It serves as a reference line along which the parabola's structure is symmetrical.

• In our specific example with \(x^2 = 4y\), the axis of symmetry is the line \(x = 0\).

This line is particularly useful in parabola plotting, as it offers a guide where the shape switches direction. Additionally, knowing its position can assist in understanding and predicting the reflective properties and behavior of the parabola.