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.

  1. 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:

3y218y+12x+3=03y^2 - 18y + 12x + 3 = 0

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: (yk)2=4p(xh)(y - k)^2 = 4p(x - h) where (h,k)(h, k) is the vertex, and pp determines the distance between the vertex and the focus.

Given equation:

3y218y+12x+3=03y^2 - 18y + 12x + 3 = 0 Let's complete the square for the yy-terms.

  1. Move the xx-terms to the other side: 3y218y=12x33y^2 - 18y = -12x - 3

  2. Factor out the 3 from the yy-terms: 3(y26y)=12x33(y^2 - 6y) = -12x - 3

  3. Complete the square inside the parentheses. To complete the square, take half of the coefficient of yy, which is 6-6, and square it: (6/2)2=9(-6/2)^2 = 9. 3(y26y+9)=12x3+273(y^2 - 6y + 9) = -12x - 3 + 27 Now, simplify: 3(y3)2=12x+243(y - 3)^2 = -12x + 24

  4. Divide both sides by 3 to simplify: (y3)2=4(x2)(y - 3)^2 = -4(x - 2) This is the standard form of the equation.

Step 2: Identify the properties

  1. Standard Form: (y3)2=4(x2)(y - 3)^2 = -4(x - 2)

  2. Opening: The negative sign in front of the xx-term (4(x2)-4(x - 2)) indicates that the parabola opens to the left.

  3. Vertex: The vertex is (h,k)(h, k). From the equation, (h,k)=(2,3)(h, k) = (2, 3).

  4. Distance between the vertex and focus: In the equation (y3)2=4p(x2)(y - 3)^2 = 4p(x - 2), 4p=44p = -4, so p=1p = -1. The distance between the vertex and the focus is p=1|p| = 1.

  5. Focus: Since the parabola opens to the left, the focus is to the left of the vertex by 1 unit. So, the focus is: (h1,k)=(21,3)=(1,3)(h - 1, k) = (2 - 1, 3) = (1, 3)

  6. Equation of the Axis of Symmetry: The axis of symmetry is the horizontal line through the vertex, so: y=3y = 3

  7. Length of the Latus Rectum: The length of the latus rectum is given by 4p=4|4p| = 4.

  8. 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 (y=3y = 3), the endpoints of the latus rectum are 2 units above and below the focus. The endpoints are: (1,3+2)=(1,5)and(1,32)=(1,1)(1, 3 + 2) = (1, 5) \quad \text{and} \quad (1, 3 - 2) = (1, 1)

  9. Equation of the Directrix: The directrix is a vertical line pp units to the right of the vertex (opposite direction of the focus). So, the equation of the directrix is: x=2+1=3x = 2 + 1 = 3

Summary of properties:

  • Standard Form: (y3)2=4(x2)(y - 3)^2 = -4(x - 2)
  • Opening: Left
  • Vertex: (2,3)(2, 3)
  • Distance between the vertex and focus: 1 unit
  • Focus: (1,3)(1, 3)
  • Axis of Symmetry: y=3y = 3
  • Length of Latus Rectum: 4 units
  • Endpoints of Latus Rectum: (1,5)(1, 5) and (1,1)(1, 1)
  • Directrix: x=3x = 3

Let me know if you need details or have any questions!

Here are some related questions:

  1. How do you graph parabolas with vertical axes?
  2. What is the significance of the directrix in a parabola?
  3. How can you determine if a parabola opens up, down, left, or right from its equation?
  4. What is the relationship between the vertex and focus in a parabola?
  5. 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