Problem 13 - Precalculus: Mathematics for Calculus

Question

An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. $$y^{2}=-24 x$$

Step-by-Step Solution

Answer

Focus at \((-6, 0)\); Directrix: \(x=6\); Focal diameter: 24.

Step 1: Identify the form of the equation

The given equation is \(y^2 = -24x\). This equation is of the form \((y-k)^2 = 4p(x-h)\), which represents a horizontally oriented parabola.

Step 2: Determine the vertex

For the equation \((y-k)^2 = 4p(x-h)\), the vertex is \((h,k)\). Since there is no \(h\) or \(k\) in the given equation \(y^2 = -24x\), we can deduce that \(h = 0\) and \(k = 0\). Thus, the vertex of the parabola is at \((0,0)\).

Step 3: Calculate the value of \(p\)

For the equation \(y^2 = -24x\), compare it with the standard form \((y-k)^2 = 4p(x-h)\). We have \(4p = -24\), which implies \(p = -6\).

Step 4: Determine the focus

The focus of a horizontal parabola \((y-k)^2 = 4p(x-h)\) is \((h+p, k)\). With \(h=0\), \(k=0\), and \(p=-6\), the focus is \((-6, 0)\).

Step 5: Find the equation of the directrix

The directrix of the parabola \((y-k)^2 = 4p(x-h)\) can be found using \(x = h-p\). With \(h = 0\) and \(p = -6\), the directrix is \(x = 6\).

Step 6: Calculate the focal diameter

The focal diameter is the absolute value of \(4p\). Since \(4p = -24\), the absolute value is \(24\). Thus, the focal diameter is \(24\).

Step 7: Sketch the graph

Draw the parabola with vertex at \((0,0)\), opening to the left since \(p=-6\) is negative. Plot the focus at \((-6, 0)\) and draw the directrix as a vertical line at \(x=6\). Include the focal diameter spanning \(24\) units across the width of the parabola.

Key Concepts

Focus

The focus is a pivotal point in understanding the properties of a parabola. For a parabola described by the equation \((y-k)^2 = 4p(x-h)\), the focus acts as a specific point inside the curve, which the parabola "curves around". In the context of our exercise, the parabola is defined by \(y^2 = -24x\), which is a horizontally oriented parabola.

• **Focus Calculation:** For horizontally oriented parabolas, the focus can be found at the coordinates \((h+p, k)\).
• **Given Equation Characteristics:** From the given equation \(y^2 = -24x\), we know the parameters \(h = 0\), \(k = 0\), and \(p = -6\).
• **Resulting Focus:** Plugging these into the formula, the focus of this parabola is located at \((-6, 0)\). This point is specifically where the distances to the curved edge are uniformly spread across the parabola's width.

Understanding the focus helps visualize how the parabola is shaped and oriented. It gives insight into the symmetry and the direction in which the parabola opens.

Directrix

The directrix of a parabola is a line that helps define its shape and orientation, acting in balance with the focus.
The directrix is essentially a guideline for the parabola, showing how far and in what direction the parabola reaches outward. For the equation \((y-k)^2 = 4p(x-h)\), the line of the directrix is expressed as \(x = h-p\).

• **Directrix Calculation:** With \(h = 0\) and \(p = -6\), the directrix is found using \(x = h-p = 0-(-6)\).
• **Line Equation:** Therefore, the equation for the directrix in our exercise is \(x = 6\), which is a vertical line.

The directrix effectively acts like a boundary that the parabola will never actually touch. When plotting it, imagine a line that parallels the path of the parabola opening, serving as an opposing stretch to the focus. This invites a visual symmetry essential for envisioning the parabola's curve.

Focal Diameter

The focal diameter, also known as the latus rectum, offers us a numerical value that expresses the width of the parabola at the level of the focus. It is determined by the length of the line segment that passes through the focus and is perpendicular to the axis of symmetry. This segment is entirely contained within the parabola.

• **Focal Diameter Formula:** It is calculated as the absolute value of \(4p\).
• **Given Equation Calculation:** For the given standard form \(y^2 = -24x\), \(4p = -24\).
• **Resulting Value:** Taking the absolute value gives us \(\text{Focal Diameter} = 24\).

The focal diameter directly shows how "wide" the parabola appears at its most significant span across its curve direction. This measure is crucial for sketching the parabola accurately, helping students visualize how the parabola opens and how it compares relative to other parabolas.