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.