Math Problem Statement
I. Graphing Parabolas: Graph the following equations in a rectangular coordinate plane. Then, identify the required properties being asked in each item.
- 3y^2 - 18y + 12 x + 3 = 0 Standard Form: Opening: Vertex: Distance between the vertex and focus: Focus: Equation of Axis of Symmetry: Length of Lactus Rectum: Endpoints of Latus Rectum: Equation of Directrix:
Solution
To graph and analyze the parabola given by the equation:
we need to rewrite it into a standard form of a parabola and then extract the required properties.
Step 1: Rewrite the equation in standard form
The standard form of a parabola with a horizontal axis is: where is the vertex, and determines the distance between the vertex and the focus.
Given equation:
Let's complete the square for the -terms.
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Move the -terms to the other side:
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Factor out the 3 from the -terms:
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Complete the square inside the parentheses. To complete the square, take half of the coefficient of , which is , and square it: . Now, simplify:
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Divide both sides by 3 to simplify: This is the standard form of the equation.
Step 2: Identify the properties
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Standard Form:
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Opening: The negative sign in front of the -term () indicates that the parabola opens to the left.
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Vertex: The vertex is . From the equation, .
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Distance between the vertex and focus: In the equation , , so . The distance between the vertex and the focus is .
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Focus: Since the parabola opens to the left, the focus is to the left of the vertex by 1 unit. So, the focus is:
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Equation of the Axis of Symmetry: The axis of symmetry is the horizontal line through the vertex, so:
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Length of the Latus Rectum: The length of the latus rectum is given by .
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Endpoints of the Latus Rectum: The latus rectum is perpendicular to the axis of symmetry and passes through the focus. Since the axis of symmetry is horizontal (), the endpoints of the latus rectum are 2 units above and below the focus. The endpoints are:
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Equation of the Directrix: The directrix is a vertical line units to the right of the vertex (opposite direction of the focus). So, the equation of the directrix is:
Summary of properties:
- Standard Form:
- Opening: Left
- Vertex:
- Distance between the vertex and focus: 1 unit
- Focus:
- Axis of Symmetry:
- Length of Latus Rectum: 4 units
- Endpoints of Latus Rectum: and
- Directrix:
Let me know if you need details or have any questions!
Here are some related questions:
- How do you graph parabolas with vertical axes?
- What is the significance of the directrix in a parabola?
- How can you determine if a parabola opens up, down, left, or right from its equation?
- What is the relationship between the vertex and focus in a parabola?
- How do the endpoints of the latus rectum relate to the focus of a parabola?
Tip: When converting a quadratic equation into standard form, always complete the square carefully to ensure accuracy.
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Math Problem Analysis
Mathematical Concepts
Algebra
Analytic Geometry
Parabolas
Formulas
(y - k)^2 = 4p(x - h)
Distance between vertex and focus = |p|
Axis of symmetry formula: y = k
Latus Rectum length = |4p|
Theorems
Completing the Square
Properties of Parabolas
Focus-Directrix Definition of a Parabola
Suitable Grade Level
Grade 10-12
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