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Questions and Answers
What defines a parabola in geometric terms?
What defines a parabola in geometric terms?
- The set of points equidistant from a point and a straight line. (correct)
- The locus of points equidistant from two parallel lines.
- The set of points equidistant from two foci.
- The intersection of a cone with a plane intersecting both nappes.
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?
- It opens upward.
- It opens to the right.
- It opens to the left. (correct)
- It opens downward.
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?
- It opens upward.
- It opens leftward.
- It opens to the right.
- It opens downward. (correct)
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)?
Which of the following describes the axis of symmetry of a parabola?
Which of the following describes the axis of symmetry of a parabola?
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 $y^2 = 6x$?
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 $x^2 = 4py$?
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?
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?
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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