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Questions and Answers
What defines a parabola in geometric terms?
What defines a parabola in geometric terms?
If the equation of a parabola is given as $y^2 = -12x$, what can be determined about its orientation?
If the equation of a parabola is given as $y^2 = -12x$, what can be determined about its orientation?
In the equation $x^2 = -10y$, what does the negative value of $p$ indicate about the parabola?
In the equation $x^2 = -10y$, what does the negative value of $p$ indicate about the parabola?
In what form is the equation of a parabola expressed when the vertex is at the point (h, k) and the focus at (h + p, k)?
In what form is the equation of a parabola expressed when the vertex is at the point (h, k) and the focus at (h + p, k)?
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Which of the following describes the axis of symmetry of a parabola?
Which of the following describes the axis of symmetry of a parabola?
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What is the vertex of the parabola given by the equation $y^2 = 6x$?
What is the vertex of the parabola given by the equation $y^2 = 6x$?
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If a parabola opens upward, what can be said about the value of p in the equation $x^2 = 4py$?
If a parabola opens upward, what can be said about the value of p in the equation $x^2 = 4py$?
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What is the directrix of the parabola defined by the equation $y^2 = 4px$ for p = 2?
What is the directrix of the parabola defined by the equation $y^2 = 4px$ for p = 2?
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Which of the following parabolas has a vertex located at the origin and opens to the left?
Which of the following parabolas has a vertex located at the origin and opens to the left?
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What property defines the distance from the vertex to the focus and the vertex to the directrix in a parabola?
What property defines the distance from the vertex to the focus and the vertex to the directrix in a parabola?
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Study Notes
Definition and Characteristics
- A parabola is formed by the intersection of a plane with one nappe of a cone.
- It is defined as the set of points equidistant from a point (focus) and a line (directrix).
- The axis of symmetry is a line perpendicular to the directrix that passes through the focus.
Key Components
- Focus: A specific point where all reflective paths converge.
- Directrix: A fixed line used in the definition of a parabola.
- Vertex: The point where the axis of symmetry intersects the parabola.
Equations of Parabolas
-
Horizontal Parabola (Vertex at Origin):
- Equation: ( y^2 = 4px )
- Opens to the right if ( p > 0 ); to the left if ( p < 0 ).
-
Vertical Parabola (Vertex at Origin):
- Equation: ( x^2 = 4py )
- Opens upward if ( p > 0 ); downward if ( p < 0 ).
Vertex Form of Parabola
- Equation: ( (y - k)^2 = 4p(x - h) )
- Focus located at ( (h + p, k) ).
- Opens to the right if ( p > 0 ); to the left if ( p < 0 ).
Example Problems
- Analyze and sketch the parabolas given in the equations:
- ( y^2 = -12x )
- ( 2x^2 - 14y = 0 )
- ( y^2 = 6x )
- ( x^2 = -10y )
- ( 2y^2 = 9x )
- ( y^2 + 7x = 0 )
- ( 3x^2 - 4y = 0 )
Determining Key Features
- For each parabola, identify:
- Vertex
- Focus
- Directrix
- Axis of symmetry
- Include these components in the sketch of the graph.
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Description
Explore the fascinating world of parabolas in this quiz on conic sections. Understand their unique characteristics, definitions, and geometric properties, including focus and directrix. Test your knowledge and learn more about the shape formed from the intersection of a plane with a cone.
