Problem 36 - Precalculus with Calculus Previews

Question

Find an equation of parabola that satisfies the given conditions. Focus \((-2,3),\) vertex (-2,5)

Step-by-Step Solution

Answer

The equation of the parabola is \((x + 2)^2 = -8(y - 5)\).

Step 1: Understanding the Vertex Form

The vertex form of a parabolic equation is given by \[ (x-h)^2 = 4p(y-k) \] where \((h, k)\) is the vertex of the parabola and \(p\) is the distance from the vertex to the focus.

Step 2: Identify Given Information

The vertex of the parabola is given as \((-2, 5)\) and the focus is given as \((-2, 3)\).Since both the vertex and the focus have the same \(x\)-coordinate, the parabola opens vertically.

Step 3: Determine the Value of p

Since the parabola opens vertically and the focus is below the vertex (because 3 < 5), \( p \) is negative. The distance \( p \) is equal to \( 5 - 3 = 2 \), so \( p = -2 \).

Step 4: Write the Equation

Plug the vertex \((-2, 5)\) and \( p = -2 \) into the vertex form equation:\[(x + 2)^2 = 4(-2)(y - 5)\]This simplifies to:\[(x + 2)^2 = -8(y - 5)\].

Key Concepts

Vertex Form of Parabola

The vertex form of a parabola is a way to express the parabola's equation that makes it simple to identify its vertex. The equation is written as \[ (x-h)^2 = 4p(y-k) \], where

• \((h, k)\) is the vertex of the parabola.
• \(p\) is the distance from the vertex to the focus.

The beauty of the vertex form is how clearly it shows both the horizontal and vertical shifts of the parabola. The vertex \((h, k)\) is the point where the parabola changes direction. By simply looking at the equation, you can quickly locate the vertex by identifying \(h\) and \(k\). If you need to create or analyze the shape of a parabola, starting with the vertex form is immensely helpful.In the given problem, the vertex is \((-2, 5)\), which easily plugs into the equation to start forming the parabola’s description.

Focus of a Parabola

The focus of a parabola is a critical point used to define the curve. It is one fixed point of the parabola, alongside the directrix, that determines the set of points forming the "bowl" shape. Everything about a parabola traces back to its relation with the focus and its partner, the vertex. In our example, the focus is \((-2, 3)\). This focus is positioned beneath the vertex, which gives us information about the parabola’s orientation.Every point on a parabola is equidistant from both the focus and a line called the directrix. This unique relationship highlights why the focus is essential for finding and understanding the equation of the parabola. By knowing the focus and the vertex, we can determine vital characteristics like the direction and curvature.

Distance from Vertex to Focus

The distance from the vertex to the focus, denoted as \(p\), plays a significant role in the equation of the parabola. This distance helps us determine how "wide" or "narrow" the parabola will appear. In mathematical terms, this distance \(p\) is used directly in the equation \[(x-h)^2 = 4p(y-k)\], where the value of \(p\) affects the parabola's stretch. In our problem, the vertex is at \((-2, 5)\) and the focus is at \((-2, 3)\). The vertical distance between these points is calculated as \(5 - 3 = 2\). Since the focus lies below the vertex, \(p\) is taken as negative, giving us \(p = -2\). This signifies that the parabola opens down, indicating the direction in which it spreads.

Parabola Opens Vertically

A parabola can open either vertically or horizontally, and in this case, it opens vertically. This means that the parabola either faces upwards or downwards along the y-axis. The equation for a vertically-opening parabola is of the form \[(x-h)^2 = 4p(y-k)\]. Because the x-coordinates of the vertex and focus are the same in our exercise, the parabola must open vertically. When a parabola opens vertically, its orientation (whether it opens up or down) is determined by the sign of \(p\). If \(p\) is positive, the parabola opens upwards. Conversely, if \(p\) is negative, as in our problem, it opens downwards. This orientation helps in visualizing the curve’s shape. Knowing its direction is critical for graphing and interpreting where and how it will extend.