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Chapter 6: Problem 6

The ___________ of a parabola is the midpoint between the focus and the directrix.

Short Answer

Expert verified
The vertex is the midpoint between the focus and the directrix of a parabola.

Step by step solution

01

Defining the involved terms.

A parabola is a set of all points in a plane that are equidistant from a given point (the focus) and a given line (the directrix).
02

Link the aspects of the parabola

The point halfway between the directrix and focus, located on the axis of symmetry, is termed the vertex, and represents the 'peak' of the parabola.

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Key Concepts

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

Focus of a Parabola
The focus of a parabola is a special point that works together with the directrix to define the parabola's shape. Imagine the open end of the parabola facing up or down; the focus lies along the axis of symmetry inside the parabola. It acts as a guide, ensuring all points on the parabola are equidistant from itself and the directrix.
From a geometric viewpoint:
  • The parabola is a U-shaped curve.
  • The focus is located at a fixed point within this curve.
  • Every point on the parabola is the same distance from the focus and the directrix.
This equidistance property is fundamental in the definition of a parabola and allows us to construct and understand its properties.
Directrix of a Parabola
The directrix of a parabola is an imaginary line situated outside the parabola, opposite the focus. This line provides a boundary, such that each point on the parabola maintains equal distance to both the focus and the directrix. In simpler terms, the directrix lies perpendicular to the axis of symmetry and serves as the reference line for the parabola.
  • It is always at the same distance from the vertex of the parabola as the focus is.
  • The direction of the directrix depends on whether the parabola opens upwards, downwards, left, or right.
The relationship between the focus and the directrix is key to understanding the geometric structure and equation of a parabola.
Vertex of a Parabola
The vertex of a parabola is the 'tip' or 'peak' point where the curve changes direction. It operates as the midpoint between the focus and the directrix. It essentially acts as the turning point on the axis of symmetry.
  • In terms of a graph, the vertex is where the curve is nearest to or farthest from the directrix.
  • For a parabola that opens upwards or downwards, the vertex represents the lowest or highest point, respectively.
  • In the standard form of a parabolic equation \[ y = ax^2 + bx + c \], the vertex can be found using the formula:
    \[ x = -\frac{b}{2a} \].
Understanding the vertex is crucial because it provides insights into the parabola's orientation and minimum or maximum values.
Axis of Symmetry
The axis of symmetry is an essential line running through the vertex and perpendicular to the directrix. This axis divides the parabola into two identical mirror-image halves. For every point on one side of the axis, there is a corresponding point on the opposite side equidistant from the axis.
  • In practical terms, the axis of symmetry helps in identifying the parabola's mirror symmetry.
  • Mathematically, the equation of the axis of symmetry for a vertically oriented parabola is:
    \[ x = h \], where \( (h, k) \) is the vertex.
The axis of symmetry is crucial in calculations and graphing as it provides a reference for balancing the parabola's two halves.

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Most popular questions from this chapter

Use a graphing utility to graph the ellipse. Find the center, foci, and vertices. (Recall that it may be necessary to solve the equation for \(y\) and obtain two equations.) \(36 x^{2}+9 y^{2}+48 x-36 y-72=0\)

Find the vertex, focus, and directrix of the parabola. Use a graphing utility to graph the parabola. \(y^{2}+x+y=0\)

Find an equation of the tangent line to the parabola at the given point, and find the \(x\) -intercept of the line. \(x^{2}=2 y,(4,8)\)

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(6 x^{2}+2 y^{2}+18 x-10 y+2=0\)

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (卤7,0)\(;\) foci: (卤2,0)
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