Problem 11 - Thomas Calculus

Question

Exercises \(9-16\) give equations of parabolas. Find each parabola's focus and directrix. Then sketch the parabola. Include the focus and directrix in your sketch. $$ x^{2}=-8 y $$

Step-by-Step Solution

Answer

The parabola opens downward with focus (0,-2) and directrix y = 2.

Step 1: Identify the Standard Form

\(x^2 = -8y\) is in the form \(x^2 = 4py\) with \(4p = -8\), so \(p = -2\).

Step 2: Find the Focus

Since the parabola opens downward (\(p < 0\)), the focus is at \((0, p) = (0, -2)\).

Step 3: Find the Directrix

The directrix is \(y = -p = 2\).

Step 4: Describe the Parabola

The parabola opens downward with vertex at the origin, focus at \((0, -2)\), and directrix \(y = 2\).

Key Concepts

Focus of a Parabola

In a parabola, the focus is a special point located inside the curve. It holds a unique position where all the points on the parabola are equidistant to the focus and some other line known as the directrix. Understanding the role of the focus can help you in sketching an accurate representation of the parabola.

For the equation given in the exercise, \( x^2 = -8y \), we need to compare it with the standard form of a vertical parabola, \( x^2 = 4py \). The coefficient \(-8\) in the equation helps us find \( p \) since \( 4p = -8 \). Thus, \( p = -2 \).

The focus of a parabola given by \( x^2 = 4py \) is always at \( (0, p) \) from the vertex, which is the origin \((0, 0)\) in this case. So, for our equation, the focus is located at \( (0, -2) \). This is below the vertex, showing the direction the parabola opens.

Directrix of a Parabola

The directrix of a parabola is a fixed line that is used in conjunction with the focus to define the shape of the parabola. Every point on the parabola is equidistant to the focus and the directrix. This means the directrix plays a crucial role in shaping the path of the parabola.

In our example, with the equation \( x^2 = -8y \), the directrix complements the position of the focus (0, -2). The directrix will be a horizontal line since it is a vertical parabola. Since the parabola opens downward, the directrix will be above the vertex at the same distance as the focus is below the vertex.

Using the value of \( p = -2 \), the directrix of the parabola is the line \( y = 2 \), which is found by taking the opposite of the \( p \) distance from the vertex. Therefore, while sketching, make sure you draw this line above the vertex at \((0, 0)\).

Graphing Parabolas

Graphing parabolas involves understanding the equation and concepts of focus and directrix to plot an accurate representation of the curve. Start by identifying key components of the equation, like whether \( p \) is positive or negative, which tells you if the parabola opens up or down.

In this exercise, \( x^2 = -8y \), the negative \( 4p \) suggests an upward reflection. Plot the vertex, which is the point \((0, 0)\) in simple cases such as ours. Mark the focus at \( (0, -2) \) and the directrix at \( y = 2 \).

• Begin at the vertex, the central point from where the curve starts.
• Draw the line for the directrix above the vertex.
• Mark the focus below the vertex.

To draw the parabola, remember it should open toward the focus (downwards in this case) and be symmetric about the vertical axis through the vertex. It’s essential to ensure equidistance is maintained between any point on the parabola, the focus, and its corresponding point on the directrix. This coherent structure ensures a precise graphical representation of any given parabola.