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Questions and Answers
What does the parameter 'a' in the standard form equation of a parabola determine?
What does the parameter 'a' in the standard form equation of a parabola determine?
- The symmetry of the parabola
- The coordinates of the vertex
- The focus and directrix distance
- The direction and width of the parabola (correct)
What coordinates represent the vertex of a parabola expressed in the standard form equation?
What coordinates represent the vertex of a parabola expressed in the standard form equation?
- (0, 0)
- (k, h)
- (-h, -k)
- (h, k) (correct)
Which of the following describes a parabola that has the equation $y = -2(x - 3)^2 + 1$?
Which of the following describes a parabola that has the equation $y = -2(x - 3)^2 + 1$?
- Opens downwards and is narrow (correct)
- Opens left and is narrow
- Opens upwards and is wide
- Opens upwards and is narrow
For the horizontal parabola described by the equation $x = -3(y - 2)^2 + 5$, what is the axis of symmetry?
For the horizontal parabola described by the equation $x = -3(y - 2)^2 + 5$, what is the axis of symmetry?
What does it mean if the absolute value of 'a' is smaller than 1 in a parabola's standard form equation?
What does it mean if the absolute value of 'a' is smaller than 1 in a parabola's standard form equation?
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Study Notes
Parabola: Standard Form Equation
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Definition: A parabola is a symmetrical curve formed by the intersection of a cone with a plane parallel to its side.
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Standard Form Equation:
- The standard form of a parabola can be expressed as:
- Vertical Parabolas: ( y = a(x - h)^2 + k )
- Horizontal Parabolas: ( x = a(y - k)^2 + h )
- The standard form of a parabola can be expressed as:
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Parameters:
- ( (h, k) ): Vertex of the parabola.
- ( a ): Determines the direction and width of the parabola.
- If ( a > 0 ): Opens upwards (vertical) or rightwards (horizontal).
- If ( a < 0 ): Opens downwards (vertical) or leftwards (horizontal).
- The absolute value of ( a ) affects the "width":
- Larger |a| = narrower parabola.
- Smaller |a| = wider parabola.
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Vertex:
- The point ( (h, k) ) is the vertex, which is the maximum or minimum point of the parabola depending on the direction it opens.
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Axis of Symmetry:
- The line that vertically or horizontally bisects the parabola:
- For vertical parabolas: ( x = h ).
- For horizontal parabolas: ( y = k ).
- The line that vertically or horizontally bisects the parabola:
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Direction of Opening:
- Vertical: Opens up/down.
- Horizontal: Opens left/right.
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Focus and Directrix:
- Each parabola has a focus point and a directrix line which are related to its geometric properties.
- Distance from the vertex to the focus and vertex to the directrix is equal.
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Graphing Tips:
- Identify the vertex ( (h, k) ).
- Determine the direction of opening using the sign of ( a ).
- Plot additional points by choosing values for ( x ) (for vertical) or ( y ) (for horizontal) to find corresponding ( y ) or ( x ) values.
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Applications:
- Parabolas are used in various fields such as physics (projectile motion), engineering (reflective properties of parabolic mirrors), and mathematics (optimization problems).
Definition and Form
- A parabola is a symmetrical curve created when a cone is intersected by a plane parallel to its side.
- Standard forms include:
- Vertical parabolas: ( y = a(x - h)^2 + k )
- Horizontal parabolas: ( x = a(y - k)^2 + h )
Key Parameters
- Vertex ( (h, k) ): The peak point of the parabola, represents the maximum or minimum based on the direction of opening.
- Coefficient ( a ):
- Positive ( a > 0 ): Opens upwards (for vertical) or rightwards (for horizontal).
- Negative ( a < 0 ): Opens downwards (for vertical) or leftwards (for horizontal).
- Magnitude of ( a ):
- Larger absolute value ( |a| ) results in a narrower parabola.
- Smaller absolute value ( |a| ) leads to a wider parabola.
Symmetry and Direction
- Axis of symmetry defines the center line:
- For vertical parabolas: ( x = h )
- For horizontal parabolas: ( y = k )
- Direction of opening is contingent on the sign of ( a ):
- Vertical parabolas open either up or down.
- Horizontal parabolas open either left or right.
Focus and Directrix
- Each parabola features a focus point and a directrix line, crucial for their geometric properties.
- The distance from the vertex to the focus is equal to the distance from the vertex to the directrix.
Graphing Techniques
- Start by identifying the vertex ( (h, k) ).
- Assess the direction of opening via the sign of ( a ).
- Use chosen values for ( x ) (in vertical parabolas) or ( y ) (in horizontal parabolas) to find corresponding coordinates and plot additional points.
Applications
- Parabolas are utilized in various disciplines including:
- Physics: Modeling projectile motion.
- Engineering: Exploiting the reflective properties of parabolic mirrors.
- Mathematics: Solving optimization problems.
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