Problem 11 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the standard equation of each parabola from the given information. Assume that the vertex is at the origin. Directrix is \(y-2=0\)

Step-by-Step Solution

Answer

The standard equation is \(x^2 = -8y\).

Step 1: Determine the Direction of Opening

Since the directrix is a horizontal line given by the equation \(y = 2\), the parabola will open upwards or downwards. For a parabola with a vertex at the origin and a horizontal directrix, the parabola opens downwards if the directrix is above it.

Step 2: Calculate the Focus

The focus of a parabola is at an equal distance from the vertex as the directrix, but on the opposite side of the vertex. Given the directrix is \(y = 2\), and the vertex is at the origin \((0,0)\), the directrix is 2 units above the vertex. Therefore, the focus will be 2 units below the vertex, at \((0, -2)\).

Step 3: Write the Standard Equation

For a parabola opening downwards with the vertex at the origin, the standard form is \(x^2 = -4py\), where \(p\) is the distance from the vertex to the focus (negative because it opens downwards). Here, \(p = 2\), so the equation is \(x^2 = -4(2)y = -8y\).

Key Concepts

Understanding the Vertex of a Parabola

In the context of a parabola, the vertex is a key point that determines its shape and position. Typically, the vertex is either the highest or lowest point, depending on the direction in which the parabola opens.

• If the vertex is at the origin \((0,0)\), it simplifies calculations and understanding, as the vertex acts as a central anchor point.
• For upward or downward-opening parabolas, the vertex indicates the minimum or maximum value.
• A parabola opening to the sides would have a vertex that acts as a center point horizontally.

Understanding how the vertex affects the parabola's opening helps in determining its equation and the position of other elements, like the focus and directrix. With the vertex at the origin in this case, it keeps things simple.

Explaining the Directrix

The directrix of a parabola is a fixed line used to define its curvature along with the focus. Together, they help determine the parabola's shape and orientation.

• The directrix is located at a specific distance from the vertex along the axis of symmetry.
• For the parabola in this exercise, the directrix is given by the equation \(y = 2\).
• This horizontal line indicates the parabola will open either upwards or downwards.

The position of the directrix, being perpendicular to the axis of symmetry, helps in understanding the opening of the parabola. If the directrix is above the vertex at the origin, it implies that the parabola opens downwards towards the negative y-axis.

Role of the Focus in a Parabola

The focus of a parabola is a special point located inside the curve, which, along with the directrix, defines the parabola's properties and orientation.

• The focus is located the same distance from the vertex as the directrix but on the opposite side.
• For this parabola, with the directrix at \(y = 2\), the focus is found at \(0, -2\).
• The focus is a critical point where all reflected lines parallel to the parabola's opening direction converge.

In parabolas that open upwards or downwards, the focus is either below or above the vertex, affecting which way the parabola opens. Understanding the position of the focus helps in writing the standard equation.

Deriving the Standard Equation

The standard equation of a parabola efficiently represents its algebraic form. For a parabola opening vertically and with the vertex at the origin, the equation is typically written as:

1. \(x^2 = 4py\) for upward opening.
2. \(x^2 = -4py\) for downward opening.

In this task:

• The parabola opens downward since the directrix is above the vertex.
• \(p\), the distance from vertex to focus, is 2, but because the parabola opens downwards, we use -2 for calculations.

Thus, the equation becomes \(x^2 = -8y\), illustrating how well-defined points and lines like vertex, focus, and directrix play a role in geometrically shaping the parabola and presenting it mathematically through its standard form.