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Chapter 10: Problem 35

Find the equation of the given conic. Parabola with vertex (2,3) and focus (2,5).

Short Answer

Expert verified
The equation of the parabola is \( (x-2)^2 = 8(y-3) \).

Step by step solution

01

Determine the Direction of the Parabola

Since the vertex (2,3) and the focus (2,5) have the same x-coordinate, the parabola opens vertically. Additionally, because the focus is above the vertex, the parabola opens upwards.
02

Calculate the Distance p

The distance between the vertex and the focus is the parameter \( p \). Here, \( p = 5 - 3 = 2 \). This means \( p = 2 \) in the formula of the parabola.
03

Use the Standard Form of the Parabola

For a parabola that opens vertically, the standard form is \( (x-h)^2 = 4p(y-k) \). The vertex is \( (h, k) = (2, 3) \). Substituting in these values gives \( (x-2)^2 = 8(y-3) \).
04

Simplify the Equation

The equation already simplifies to \( (x-2)^2 = 8(y-3) \), which is neat and ready to use. Here, we've used \( 4p = 8 \) as \( p = 2 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertex
The vertex of a parabola is a key point that helps define its shape and direction. Imagine the vertex as the tip or "corner" point where the parabola turns. In this example, the given vertex is (2, 3). This means that on a graph, the parabola changes its direction or reaches its minimum point at this location.
Understanding the vertex's role is vital, as it helps in drafting the parabola's graph and finding its equation.
  • The vertex indicates the point of symmetry for the parabola.
  • It's essential in the process of determining the equation of the parabola, especially when converting it into its standard form.
Always keep in mind, when the parabola opens upwards or downwards, the vertex marks the lowest or highest point on the curve, respectively.
Focus
The focus of a parabola is another special point related to its geometry. Think of the focus as a point that determines how "wide" or "narrow" the parabola opens. In this exercise, the focus is located at (2, 5).
The focus lies along the axis of symmetry of the parabola, and in this case, since the x-coordinates of the vertex and focus are the same, the parabola opens vertically.
  • The focus and vertex together help in deriving the direction in which the parabola opens, which is upwards in this example as the focus is above the vertex.
  • The distance between the focus and the vertex is crucial, defined as the parameter "p," which influences the calculation within the standard form of the parabola.
Remember, the closer the focus is to the vertex, the sharper or "tighter" the parabola appears.
Standard Form
The standard form of a parabola's equation is especially helpful for understanding its properties and graph. For parabolas that open vertically, like in our example, the standard form is given by the equation \((x-h)^2 = 4p(y-k)\). Here, \((h, k)\) denotes the vertex of the parabola.
In this article's example, with the vertex at (2, 3) and a focus at (2, 5), the distance \(p\) is calculated as 2. Hence, the equation becomes \((x-2)^2 = 8(y-3)\).
  • In this equation, \(4p\) was calculated as 8 because \(p = 2\).
  • The standard form makes it easy to identify the vertex and calculate how the parabola opens, based on the sign and position of \(p\).
Using the standard form is a perfect way to derive everything about a parabola, from its vertex location to its general shape.

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Most popular questions from this chapter

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