Problem 39 - Precalculus: Mathematics for Calculus

Question

Find an equation for the conic section with the given properties. The parabola with vertex \(V(-3,5)\) and directrix \(y=2\)

Step-by-Step Solution

Answer

The equation of the parabola is \((x + 3)^2 = 12(y - 5)\).

Step 1: Identify the form of the parabola

A parabola with a horizontal or vertical axis can be represented in the form \[ y = a(x - h)^2 + k \] or \[ x = a(y - k)^2 + h \], where \((h, k)\) is the vertex of the parabola. Given that the vertex is \(V(-3, 5)\) and the directrix is a horizontal line \(y = 2\), the parabola opens vertically.

Step 2: Calculate the focus

The distance between the vertex and the directrix is the same as the distance between the vertex and the focus. In this case, the distance is \(5 - 2 = 3\). Since the vertex is at \(y = 5\) and the directrix is at \(y = 2\), the focus will be at 3 units above the vertex. Hence, the focus is at \((-3, 8)\).

Step 3: Determine the value of \(a\)

The formula for a parabola is \( (x - h)^2 = 4p(y - k) \), where \(p\) is the distance between the vertex and the focus (or directrix). We calculated \(p = 3\), so \(4p = 12\). Thus, the equation is \( (x + 3)^2 = 12(y - 5) \).

Step 4: Simplify the equation

Expanding the parabola equation, we write \[(x + 3)^2 = 12(y - 5)\] This can be expanded but this form is a standard form to represent the parabola as requested in vertex form.

Key Concepts

vertex

The vertex of a parabola is a key component that determines its shape and position. Imagine it as the ‘tip’ of the parabola where it changes direction.
This point is crucial because it is always equidistant to the focus and the directrix, maintaining the unique property of the parabola being equidistant to both.

• In the exercise, the vertex is given as \(-3, 5\), demonstrating how to position the parabola on a coordinate plane.
• The vertex \(h, k\) helps define the parabola's standard form equation, particularly indicating the amounts of horizontal and vertical shifts applied to the basic parabola shape \(y = x^2\) or \(x = y^2\).

Understanding the vertex's role and its relationship with the rest of the components allows for a better grip on crafting the equation of the parabola.

directrix

The directrix is an essential line, assisting in shaping and defining a parabola's curve. Unlike the vertex, the directrix does not intersect the parabola. Instead, it serves as a guidepost.

• In our example, the directrix is a horizontal line \( y = 2 \).
• This line is a measure of how far the parabola should bend or curve in response to its proximity to the directrix.

The distance from any point on the parabola to the directrix is equal to its distance to the focus. This property keeps the parabola balanced in its symmetrical form. When calculating or graphing parabolas, knowing the position and equation of the directrix is as vital as knowing where the vertex lies.

focus

The focus of a parabola is a special point inside the curve. Much like a beacon, its position helps dictate how the parabola stretches outward. The focus and the directrix work together to define the parabola's shape.

• For our exercise, the focus is calculated to be at \(-3, 8\), just above the vertex reflecting its vertical opening.
• Being equidistant from the vertex along with the directrix, ensures the parabola's reflective and consistent nature.

Understanding the placement of the focus can help in graphing the parabola and comprehending the physical implications, like satellite dishes or car headlights functioning under similar principles.

equation of parabola

Forming the equation of a parabola involves taking the vertex, focus, and directrix into account. The general form for a parabola with a vertical axis is \( (x - h)^2 = 4p(y - k) \), whereas a horizontally oriented parabola uses the form \( (y - k)^2 = 4p(x - h) \).

• Here, deriving from the information \((-3, 5)\) as the vertex and \(y = 2\) as the directrix, we identify it as a vertically opening parabola.
• Using the calculated distance \(p = 3\), signifies how far the vertex is from the focus and the directrix (here \(4p = 12\)).
• This concludes our equation with: \((x + 3)^2 = 12(y - 5)\).

Each component plays a part in forming the precise parabolic curve, exemplifying these mathematical relationships can direct most transformations or constructions involving parabolas.