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

Focus: \((0, -2)\); Directrix: \(y = 2\).

Step 1: Recognizing the Parabola Equation

The given equation is \(x^2 = -8y\). This can be compared with the standard form of a parabola that opens down or up, which is \(x^2 = 4py\). We can immediately determine that \(4p = -8\).

Step 2: Finding the Value of \(p\)

From \(4p = -8\), solve for \(p\) by dividing both sides by 4: \(p = -2\). This tells us the parabola opens downwards since \(p\) is negative.

Step 3: Determining the Focus

The focus of the parabola can be determined from the vertex (which is at the origin, \((0, 0)\) in this case, because there is no transformation indicated in the equation) by moving \(p\) units in the direction the parabola opens. Since \(p = -2\) and the parabola opens downward, the focus is at \((0, 0 + (-2)) = (0, -2)\).

Step 4: Determining the Directrix

The directrix is a line that is \(p\) units away from the vertex in the opposite direction to how the parabola opens. Since \(p = -2\), the directrix is \(y = 0 - (-2) = y = 2\). Thus, the directrix is the horizontal line \(y = 2\).

Step 5: Sketching the Parabola

To sketch the parabola, plot the vertex at the origin \((0, 0)\), the focus at \((0, -2)\), and draw the directrix line \(y = 2\). Draw a symmetric parabola opening downwards from the vertex; it should be equidistant from the directrix and focus.

Key Concepts

Focus of a Parabola

The focus of a parabola is a crucial point that helps in defining the curve's shape. It is located inside the parabola and influences how the parabola is drawn. To determine the focus, you begin at the vertex, which is the point where the parabola changes direction. For a parabola in the form of \(x^2 = 4py\), the vertex is located at \((0, 0)\) if there are no transformations indicated by the equation.

The focus is found by moving a distance \(p\) from the vertex in the direction the parabola opens. In our specific example, the equation \(x^2 = -8y\) tells us \(p = -2\). Since the parabola opens downward, you move 2 units downward from the vertex, placing the focus at \((0, -2)\).

• The focus creates a focal point used to reflect and direct lines (known as rays) parallel to the axis of symmetry.
• All points on the parabola are equidistant to the focus and the directrix, a property that characterizes the parabola.

Even without seeing the entire curve, knowing the focus gives us critical insight into its structure and direction.

Directrix of a Parabola

The directrix of a parabola acts like its guiding line. It is a fixed line that helps form the parabola by acting as a reference from which distances are equally measured against the focus. For the equation \(x^2 = 4py\), the directrix is located at the line \(y = -p\) when the parabola opens upwards or downwards.

In our case, looking at \(x^2 = -8y\) where \(p = -2\), the directrix lies horizontally at \(y = 2\). This line is 2 units above the vertex, since the parabola opens downwards.

• The directrix ensures the parabola maintains its shape since each point on the parabola is equidistant to both the focus and the directrix.
• It serves as an imaginary boundary, and as you plot the parabola, ensure the curve stays equally distant between the directrix and the focus.

Working with the directrix not only allows you to sketch the curve accurately but also strengthens your understanding of how parabolas behave geometrically.

Standard Form of a Parabola

Understanding the standard form of a parabola is essential for solving any parabola-related problem. The standard form for parabolas opening up or down is \(x^2 = 4py\), while for parabolas opening left or right, it would be \(y^2 = 4px\). This form helps in identifying \(p\), the distance from the vertex to the focus and the vertex to the directrix.

When we say a parabola is in standard form, it removes any transformations like translations or rotations, meaning the vertex is at the origin \((0, 0)\). Thus, equations can easily be matched and solved using a systematic approach:

• Identify \(4p\) from the equation and solve for \(p\), which tells you how far the focus and directrix are from the vertex.
• Use the sign of \(p\) to determine the direction the parabola opens - positive opens up, negative opens down.

Recognizing a parabola in its standard form enables quick calculation of key components like focus and directrix, thereby simplifying solving and graphing tasks effortlessly.