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9.1 – From Lines to Conic Sections

Key Terms

  • Y-intercept – A point where the graph of a function crosses the y-axis. A function has at most one y-intercept.
    • The y-intercept of the line with equation y = mx + b is the point (0, b).
  • Circle – The set of all points on a plane that are the same distance from a single point, called its center.
    • A circle is a special type of ellipse.
  • Cone – A cone is a three-dimensional figure with one vertex and a circular base.
  • Conic Section – A curve that is the intersection of a right circular cone (usually a double cone) and a plane.
    • The main types of conic sections are circles, ellipses, hyperbolas, and parabolas.
  • Cross-Sections – The two-dimensional shapes that, when stacked, form a solid geometric figure.
  • Ellipse – The set of points in a plane for which the distance from the point to one focus, plus the distance from the point to the other focus, is a constant.
    • An ellipse is a type of conic section.
  • Hyperbola – The set of points in a plane that are related to two fixed points (each of which is called a focus) by this relationship:
    • The distance from the point to the first focus, minus the distance from the point to the other focus, is a constant. A hyperbola has two matching halves, called branches.
    • A hyperbola is a type of conic section.
  • Linear Equation – An equation whose graph is a straight line.
    • A linear equation can be written in the form y = mx + b.
  • Nappe – Half of a right circular cone.
  • Parabola – The set of points in a plane that are related to a given point (the focus) and a given line (the directrix) by this relationship:
    • The distance from any point on the parabola to the focus is the same as the distance from that point to the directrix.
    • A parabola is a type of conic section.
  • Right Circular Cone – A figure formed by the intersection of two ordinary, hollow cones with circular bases and axes that are perpendicular to each base.
    • The bases of a right circular cone are parallel.
  • Vertex of a Right Circular Cone – The point where the two nappes of a right circular cone meet.

 

Review

 y-Intercept
  • When the y-intercept is changed:
    • As the y-intercept increases, the graph of the line shifts up.
    • As the y-intercept decreases, the graph of the line shifts down.
 Slope Steepness
  • As a line becomes steeper, the value of its slope gets bigger.
  • When the value of the slope gets smaller, the graph of a line gets less steep.
  • Slope steepness is based on the absolute value of the coefficient of x.
    • Negative and positive signs do not matter.  For instance, -24x is steeper than 2x.
Lines with Negative Slopes
  • The line goes down from left to right.
  • As the absolute value of the negative slope gets bigger, the graph of the line gets steeper.
  • As the absolute value of the negative slope gets smaller, the graph of the line gets less steep.

Notes

 Conic Sections
  • Figures defined by the intersection of a cone and a plane.
  • A point, line, and pair of intersecting lines are special types of conic sections.
  • Conic Sections are:
    • two-dimensional.
    • “slices” of a cone.
    • formed by a plane intersecting a cone.
GeoB - 9.1 Right Circular ConeGeoB - 9.1 Right Circular Cone Parts

 

Circles
  •  A circle is produced when you slice a cone with a plane that is parallel to the base of the cone.

GeoB - 9.1 Right Circular Cone Circle

Ellipses
  • An ellipse is produced when you slice a cone with a plane that passes through only one nappe of the cone but is not parallel to an edge of the cone.
  • How to Make an Ellipse
    • Use a plane to intersect a cone.
    • Make sure the plane goes through only one nappe.
    • Make sure the plane is not parallel to an edge of the cone.

GeoB - 9.1 Right Circular Cone Elliipse

Parabolas
  • Given a right circular cone, a plane that intersects the cone (not at the vertex) and is parallel to its edge will always result in a parabola, regardless of the shape of the cone.

GeoB - 9.1 Right Circular Cone Parabola


  •  Comparing Parabolas and Ellipses

GeoB - 9.1 Parabola Ellipse

Hyperbola
  • If both nappes of a right circular cone are intersected by a plane that does not pass through the vertex of the cone, the resulting curve will be a hyperbola.

GeoB - 9.1 Right Circular Cone Hyperbola

Examples

  • Ex 1. Which equation has the steepest graph?
    • a. y=\frac{3}{4}-9
    • b. y=10x-5
    • c. y=2x+8
    • d. y=-14x+1
      • Answer: d. Because 14 is the largest absolute value of all of the slopes listed. It doesn’t matter that the slope goes down from left to right (negative). It is still the steepest!
  • Ex 2. If y = x – 6 were changed to y = x + 8, how would the graph of the new function compare with the first one?
    • Answer: It would be shifted up.
  • Ex 3. If y=5x-2 were changed to y=\frac{3}{4}x-2, how would the graph of the new function compare with the first one?
    • Answer: It would be less steep.

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