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:

To graph and analyze the parabola given by the equation:

\[
3y^2 - 18y + 12x + 3 = 0
\]

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

#### Given equation:
\[
3y^2 - 18y + 12x + 3 = 0
\]
Let's complete the square for the \(y\)-terms.

1. Move the \(x\)-terms to the other side:
   \[
   3y^2 - 18y = -12x - 3
   \]

2. Factor out the 3 from the \(y\)-terms:
   \[
   3(y^2 - 6y) = -12x - 3
   \]

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

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

### Step 2: Identify the properties

1. **Standard Form:**
   \[
   (y - 3)^2 = -4(x - 2)
   \]

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

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

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

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:
   \[
   (h - 1, k) = (2 - 1, 3) = (1, 3)
   \]

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

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

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

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

### Summary of properties:

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

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.

Graph the following equations in a rectangular coordinate plane. Then, identify the required properties being asked in each item.

1. 3y^2 - 18y + 12x + 3 = 0
Standard Form:
Opening:
Vertex:
Distance between the vertex and focus:
Focus:
Equation of Axis of Symmetry:
Length of Latus Rectum:
Endpoints of Latus Rectum:
Equation of Directrix:

Learn how to graph and analyze the parabola given by the equation 3y^2 - 18y + 12x + 3 = 0. This problem involves rewriting the equation in standard form and identifying key properties like the vertex, focus, and directrix. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation