Algebra 2 Notes
Algebra 2 Notes Name: _________
Parabolas
DAY ONE:
In Chapter 5, you learned that the graph of a quadratic function is a parabola. Because a parabola is a conic section, it can also be defined in terms of distance.
A parabola is the set of all points [pic] in a plane that are equal ____________________ from both a fixed point, the ____________________, and a fixed line, the ____________________. A parabola has an axis of ____________________ which is ____________________ to its directrix and that passes through its ____________________. The vertex of a parabola is the midpoint of the ____________________ connecting the focus and the directrix.
Previously, you have graphed parabolas with vertical axes of symmetry the open upward or downward. Parabolas may also have ____________________ axes of symmetry and may open left or right.
The equations of parabolas use the parameter [pic]. The [pic] gives the distance from the vertex to BOTH the focus and the directrix.
|Standard Form for the Equation of a Parabola with a Vertex at [pic] |
|Axis of Symmetry |Horizontal |Vertical |
| |[pic] |[pic] |
|Equation |[pic] |[pic] |
|Direction |Opens right if [pic] |Opens upward if [pic] |
| |Opens left if [pic] |Opens downward if [pic] |
|Focus |[pic] |[pic] |
|Directrix |[pic] |[pic] |
|Graph |[pic] |[pic] |
Example 1: Find the requested information for each parabola. Then graph.
|a. [pic] |b. [pic] |
|[pic] |[pic] |
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Example 2: Find the requested information. Then use it to write the parabola’s equation in standard form.
|a. |
|Direction it opens: ________________ |
|Vertex: [pic] |
|Value of [pic]: _________ |
|Axis of Symmetry: [pic] |
|Focus: [pic] |
|Directrix: [pic] |
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|Equation in Standard Form: |
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|______________________________ |
|b. |
|Direction it opens: ________________ |
|Vertex: [pic] |
|Value of [pic]: _________ |
|Axis of Symmetry: [pic] |
|Focus: [pic] |
|Directrix: [pic] |
| |
|Equation in Standard Form: |
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|______________________________ |
Example 3: Find the standard form of each parabola by completing the square.
|a. [pic] |b. [pic] |
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DAY TWO:
Example 4: Write the equation of each parabola in standard form, based on its description. Then given the domain and range. A sketch may be helpful.
|a. vertex at [pic] and directrix at [pic] |b. vertex at [pic] and focus at [pic] |
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|Direction it opens: ________________ |Direction it opens: ________________ |
|Value of [pic]: _________ |Value of [pic]: _________ |
|Equation in Standard Form: |Equation in Standard Form: |
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|______________________________ |______________________________ |
|Domain: _____________ |Domain: _____________ |
|Range: ______________ |Range: ______________ |
|c. focus at [pic] and directrix at [pic] |d. focus at [pic] and directrix at [pic] |
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|Direction it opens: ________________ |Direction it opens: ________________ |
|Value of [pic]: _________ |Value of [pic]: _________ |
|Equation in Standard Form: |Equation in Standard Form: |
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|______________________________ |______________________________ |
|Domain: _____________ |Domain: _____________ |
|Range: ______________ |Range: ______________ |
-----------------------
Direction it opens: ________________
Vertex: [pic]
Value of [pic]: _________
Axis of Symmetry: [pic]
Focus: [pic]
Directrix: [pic]
Direction it opens: ________________
Vertex: [pic]
Value of [pic]: _________
Axis of Symmetry: [pic]
Focus: [pic]
Directrix: [pic]
[pic]
[pic]
................
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