Problem 71 - A Graphical Approach to Precalculus with Limits

Question

Give the focus, directrix, and axis of each parabola. $$y^{2}=\frac{1}{16} x$$

Step-by-Step Solution

Answer

Focus: \((\frac{1}{64}, 0)\), Directrix: \(x = -\frac{1}{64}\), Axis: \(y = 0\).

Step 1: Identify the Form

First, recognize that the given equation, \(y^2 = \frac{1}{16} x\), is in the form \(y^2 = 4px\). This indicates that the parabola opens sideways (to the right). The comparison yields \(4p = \frac{1}{16}\).

Step 2: Calculate the Value of p

From the equation \(4p = \frac{1}{16}\), solve for \(p\) by dividing both sides by 4:\[ p = \frac{1}{16} \div 4 = \frac{1}{64}. \]

Step 3: Determine the Focus

The focus of a parabola in the form \(y^2 = 4px\) is located at \((p, 0)\). Since \(p = \frac{1}{64}\), the focus is at \(\left(\frac{1}{64}, 0\right)\).

Step 4: Find the Directrix

The directrix is a vertical line at \(x = -p\). Using \(p = \frac{1}{64}\), the directrix is \(x = -\frac{1}{64}\).

Step 5: Identify the Axis of Symmetry

For the equation \(y^2 = \frac{1}{16} x\), the axis of symmetry is the horizontal line \(y = 0\) since the variable \(y\) is squared.

Key Concepts

Focus of a Parabola

The focus of a parabola is a special fixed point that, along with the directrix, helps to define the parabola's unique shape. Consider the equation provided in the problem: \(y^2 = \frac{1}{16}x\). This equation is in one of the standard forms of a parabola: \(y^2 = 4px\), which means the parabola opens sideways instead of the usual up or down direction.

• For this form, the focus is located at \((p, 0)\).
• In the solution, we found that \(p = \frac{1}{64}\).
• This means the focus of the parabola is \(\left(\frac{1}{64}, 0\right)\).

Every point on the parabola is equidistant from the focus and the directrix. This geometric property makes the focus a crucial aspect in understanding and sketching parabolas.

Directrix of a Parabola

The directrix of a parabola is a straight line that serves as a reference point to help define the curve, alongside the focus. For our problem, with the equation \(y^2 = \frac{1}{16}x\), the form \(y^2 = 4px\) gives additional insight.

• The directrix is a vertical line in this case.
• It is located at \(x = -p\).
• When \(p = \frac{1}{64}\), the directrix becomes \(x = -\frac{1}{64}\).

This means that every point on the parabola maintains equal distance between the focus and this directrix line, which is one of the defining characteristics of a parabolic curve. Understanding the directrix is essential for graphing and analyzing parabolic structures.

Axis of Symmetry

The axis of symmetry of a parabola is an invisible line that divides the parabola into two mirror-image halves. This line is vital as it indicates the direction the parabola opens and gives insights into its geometric properties.In the provided equation \(y^2 = \frac{1}{16}x\), we observe that:

• The squared variable is \(y\), suggesting that the parabola opens sideways.
• The horizontal line \(y = 0\) represents the axis of symmetry.

Since this parabola opens to the right, the axis of symmetry is horizontal, and any point on the parabola has its corresponding point on the opposite side, exactly along this line. This symmetry is integral for problem-solving and graph comprehension involving parabolas.