half-plane

(noun)

One of the two parts of the coordinate plane created when a line is drawn.

Related Terms

  • boundary line

Examples of half-plane in the following topics:

  • Graphing Inequalities

    • A straight line drawn through the plane divides the plane into two half-planes, as shown in the diagram below.
    • Next, choose a test point to figure out which half-plane we need to shade in.
    • This a true statement, so shade in the half-plane containing $(0, 0). $
    • All points in the shaded half-plane above the line are solutions to this inequality.
    • The boundary line shown above divides the coordinate plane into two half-planes.
  • The Cartesian System

    • Named for "the father of analytic geometry," 17th-century French mathematician René Descartes, the Cartesian coordinate system is a 2-dimensional plane with a horizontal axis and a vertical axis.
    • Points are specified uniquely in the Cartesian plane by a pair of numerical coordinates, which are the signed distances from the point to the two axes.
    • Therefore, you move one and a half units left and two and a half units down.
    • A Cartesian plane is particularly useful for plotting a series of points that show a relationship between two variables.
  • Parabolas As Conic Sections

    • In other words, the plane is at the same angle as the outside surface of the cone.
    • The axis of symmetry is a line that is at the same angle as the cone and divides the parabola in half.
    • The vertex is the point where the plane intersects the exterior surface of the cone.
    • On a coordinate plane, parabolas are frequently encountered as graphs of quadratic functions, such as:
    • This diagram shows how a parabola is generated by the intersection of a plane with a right circular cone.
  • Ellipses

    • In mathematics, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve .
    • Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis.
    • The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, respectively.
    • The area enclosed by an ellipse is $\pi ab$, where a and b are one-half of the ellipse's major and minor axes respectively.
    • One definition of an ellipse is the intersection of a cone with an inclined plane.
  • Circles as Conic Sections

    • The set of all points in a plane that are the same distance from a given point forms a circle.
    • The set of all points in a plane that are the same distance from a given point forms a circle.
    • Radius: a line segment joining the center of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.
  • Inconsistent and Dependent Systems in Three Variables

    • Graphically, the solutions fall on a line or plane that is the intersection of three planes in space.
    • Notice that two of the planes are the same, and they intersect the third plane on a line.
    • The equations could represent three parallel planes, two parallel planes and one intersecting plane, or three planes that intersect the other two but not at the same location.
    • (b) Two of the planes are parallel and intersect with the third plane, but not with each other.
    • Two equations represent the same plane, and these intersect the third plane on a line.
  • What Are Conic Sections?

    • Conic sections can be generated by intersecting a plane with a cone.
    • Conic sections are generated by the intersection of a plane with a cone.
    • If the plane is parallel to the generating line, the conic section is a parabola.
    • If the plane is perpendicular to the axis of revolution, the conic section is a circle.
    • Each conic is determined by the angle the plane makes with the axis of the cone.
  • Types of Conic Sections

    • Depending on the angle between the plane and the cone, four different intersection shapes can be formed.
    • A parabola is formed when the plane is parallel to the surface of the cone, resulting in a U-shaped curve that lies on the plane.
    • A circle is formed when the plane is parallel to the base of the cone.
    • On a coordinate plane, the general form of the equation of the circle is
    • This happens when the plane intersects the apex of the double cone.
  • Exponential Decay

    • The time it takes for a substance (drug, radioactive nuclide, or other) to lose half of its pharmacological, physiological, biological, or radiological activity is called its half-life.
    • Then in $5$ years half the amount ($50$ pounds) remains.
    • C-13 has a half-life of 5700 years—that is, in 5700 years, half of a sample of C-13 will have converted to C-12, which represents approximately all the remaining carbon.
    • Using the graph, find that half-life.
    • Since there is 50% of the substance left after 1 year, the half-life is 1 year.
  • Introduction to Complex Numbers

    • Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
    • Thus, for example, complex number $-2+3i$ would be associated with the point $(-2,3)$ and would be plotted in the complex plane as shown below.
    • The complex number $-2+3i$ is plotted in the complex plane, $2$ to the left on the real axis, and $3$ up on the imaginary axis.
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