Problem 69 - A Graphical Approach to College Algebra

Question

Give the focus, directrix, and axis of each parabola. $$x^{2}=-\frac{1}{2} y$$

Step-by-Step Solution

Answer

Focus: \((0, -\frac{1}{8})\), Directrix: \(y = \frac{1}{8}\), Axis: \(x = 0\).

Step 1: Understanding the Standard Form of a Parabola

The equation given is not in the typical form for identifying the elements of a parabola. A parabola that opens up or down has the standard form \( (x - h)^2 = 4p(y - k) \), where \((h, k)\) is the vertex. This allows for determining the focus and directrix.

Step 2: Rewriting the Equation

The given equation is \( x^2 = -\frac{1}{2} y \). We can rewrite this to fit the standard form by expressing it in terms of \(y\): \( (x - 0)^2 = -\frac{1}{2}(y - 0) \). This indicates a parabola with its vertex at the origin \((0, 0)\).

Step 3: Identifying the Direction of the Parabola

Since the coefficient of \(y\) is negative, \( (x - 0)^2 = -\frac{1}{2}(y - 0) \), the parabola opens downwards.

Step 4: Finding the Value of p

In the standard form \( (x - h)^2 = 4p(y - k) \), we have \( 4p = -\frac{1}{2} \). Solving for \( p \) gives \( p = -\frac{1}{8} \).

Step 5: Determining the Focus

The focus of the parabola can be found using the vertex and \( p \). The formula for the focus of a vertically opening parabola is \((h, k + p)\). Here, with \((h, k) = (0,0)\) and \(p = -\frac{1}{8}\), the focus is at \( (0, -\frac{1}{8}) \).

Step 6: Identifying the Directrix

The directrix of a parabola is given by the equation \( y = k - p \). Substituting \( (h,k) = (0,0) \) and \( p = -\frac{1}{8} \), the directrix is \( y = \frac{1}{8} \).

Step 7: Determining the Axis of Symmetry

The axis of symmetry of a parabola opening up or down given by \( (x - h)^2 = 4p(y - k) \) is a vertical line passing through \(x = h\). Thus, the axis of symmetry for this parabola is \( x = 0 \).

Key Concepts

Focus

In a parabola, the focus is a unique point that plays a crucial role in defining its shape. The focus is located inside the parabola, and every point on the parabola is equidistant from the focus and the directrix, a line we'll explore next.

For the equation \[x^2 = -\frac{1}{2} y\]the vertex is at the origin \((0, 0)\).To find the focus, we use the formula \((h, k + p)\).Here, \(p\)is \(-\frac{1}{8}\),so the focus is at \((0, -\frac{1}{8})\):

• The focus is a guide for the direction of the parabola.
• It lies along the axis of symmetry.

This means all rays emanating from the focus reflect off the parabola symmetrically.

Directrix

The directrix is a horizontal line that provides a reference for constructing the parabola. In simpler terms, the directrix is an imaginary boundary—points on the parabola are equidistant from both the focus and the directrix.

For our parabola, described by the equation \(x^2 = -\frac{1}{2} y\),the directrix can be found using the formula \(y = k - p\).Given \(k = 0\)and \(p = -\frac{1}{8}\),the directrix is \( y = \frac{1}{8} \):

• The directrix is always perpendicular to the axis of symmetry.
• It helps ensure that all points on the parabola are equally spaced from it and the focus.

This line is crucial for the geometric balance of the parabola.

Axis of Symmetry

The axis of symmetry is a vertical line that runs through the vertex of the parabola, serving as a mirror that splits the parabola into two mirror-image halves. This line is essential for understanding the parabola's visual symmetry.

In the parabola given by the equation \(x^2 = -\frac{1}{2} y\),the vertex is at \((0, 0)\),so the axis of symmetry is \(x = 0\):

• Every point on the parabola is mirrored across this line.
• The focus also lies on the axis of symmetry.

The axis of symmetry helps in predicting the shape and direction of the parabola.