Problem 22 - Algebra 2

Question

Identify the focus and the directrix of the graph of each equation. $$ x=-\frac{1}{18} y^{2} $$

Step-by-Step Solution

Answer

The focus of the parabola is at (\(\frac{9}{2}\), 0) and the equation of the directrix is \(x = -\frac{9}{2}\).

Step 1: Identify the Given Equation

The equation provided is \(x = -\frac{1}{18}y^2\). Compare this with the general equation \( x = \pm \frac{1}{4a} y^2 \). Here, we see that the constant before \( y^2 \) is negative, meaning that our parabola opens to the left.

Step 2: Calculate the Value of 'a'

The coefficient of \(y^2\) in the given equation is \(-\frac{1}{18}\). Comparing this with \(\frac{1}{4a}\) in the general equation, we can set \(-\frac{1}{18} = \frac{1}{4a}\) and solve for \(a\). The solution is \(a = -\frac{1}{4}\cdot18 = -\frac{9}{2}\). The negative sign indicates that the parabola opens to the left.

Step 3: Find the Focus of the Parabola

In the parabola that opens left or right, the focus has coordinates \((h-a, k)\) where \((h, k)\) is the vertex of the parabola. In this case, since the parabola has no shifts, the vertex is at the origin (0,0), hence the focus will be at \((-a, 0)\). Substituting \(a = -\frac{9}{2}\) gives the focus as \((\frac{9}{2}, 0)\).

Step 4: Find the Directrix of the Parabola

The equation of the directrix in a parabola that opens left or right is \(x = h + a\). Since the parabola has no shifts, the vertex is (0,0) and hence the directrix will be \(x = a\). Substituting \(a = -\frac{9}{2}\) gives the directrix as \(x = -\frac{9}{2}\).

Key Concepts

Parabola Equation

A parabola is a curve where each point is equidistant from a fixed point, called the focus, and a line, called the directrix. The standard form of a parabola's equation depends on its orientation. When a parabola opens left or right, the equation takes the form:

• \( x = \pm \frac{1}{4a} y^2 \)

This is different from parabolas that open up or down, which are expressed as

• \( y = \pm \frac{1}{4a} x^2 \)

In these equations, the value \( a \) determines the width and direction of the parabola's opening. Given \( x = -\frac{1}{18} y^2 \), it fits the form for horizontal parabolas (open left or right). Since the coefficient of \( y^2 \) is negative, it indicates that the parabola opens to the left.

Vertex Form of Parabola

The vertex form of a parabola provides an easy way to identify its vertex, which is the highest or lowest point on the curve depending on its direction. For parabolas opening left or right, the vertex form is expressed as:

• \( x = a(y-k)^2 + h \)

Here, \((h, k)\) represents the vertex of the parabola. In the equation \( x = -\frac{1}{18} y^2 \), we see no \( h \) or \( k \) terms, which means the vertex is at the origin (0,0). This simplicity helps us easily determine properties related to the parabola's orientation and critical points.

Parabola Opening Direction

The direction a parabola opens depends on the coefficient of the squared term in its equation. For the equation of a horizontally oriented parabola \( x = \pm \frac{1}{4a} y^2 \), the sign of the coefficient determines its opening direction:

• If the coefficient is positive, the parabola opens to the right.
• If the coefficient is negative, the parabola opens to the left.

In our example, \( x = -\frac{1}{18} y^2 \), the negative sign before \( y^2 \) confirms the parabola opens to the left. This aspect of a parabola's graphical behavior is crucial for understanding its geometric properties, including its symmetry and intercepts.

Focus and Directrix Relationship

The focus and directrix are key elements in defining a parabola. They provide a geometric means to construct the curve. For a horizontally oriented parabola equation \( x = \pm \frac{1}{4a} y^2 \), the relationships are as follows:

• **Focus**: Located at \((h-a, k)\)
• **Directrix**: A vertical line given by \( x = h + a \)

Here, \((h, k)\) is the vertex. In the equation \( x = -\frac{1}{18} y^2 \), the vertex is at \((0,0)\), and we've calculated \( a = -\frac{9}{2} \). The focus, therefore, is \((\frac{9}{2}, 0)\), and the directrix is the line \( x = -\frac{9}{2} \). These fundamental points and line help illustrate how each point on the parabola is equidistant to the focus and the directrix.