https://file.aiquickdraw.com/mathbot/file/1728910573995bbg4f4bl.jpg

Let's analyze the two questions from your image:

### Question 1: 
The equation is \((x - 3)^2 = -4(y + 3)\).

This is a **parabola** equation, written in the form \((x - h)^2 = 4p(y - k)\), which represents a **vertical parabola** that either opens **upward** or **downward**, depending on the sign of the constant. 

- If \(4p\) is positive, the parabola opens **upward**.
- If \(4p\) is negative, the parabola opens **downward**.

In your equation, the coefficient is \(-4\), which is negative. Hence, the parabola opens **downward**.

### Answer: **Down**

---

### Question 2:
The equation is \(x^2 = 24y\).

This is also a **parabola** equation, written in the form \(x^2 = 4py\), which represents a parabola that opens either **upward** or **downward**.

- The vertex of this parabola is at \((0, 0)\), since there are no shifts in \(x\) or \(y\).
- The value \(4p = 24\) gives \(p = 6\).

  - The **focus** is at \((0, p) = (0, 6)\).
  - The equation of the **directrix** is \(y = -p = -6\).

### Summary:
- **Vertex**: \((0, 0)\)
- **Focus**: \((0, 6)\)
- **Equation of the Directrix**: \(y = -6\)

---

Would you like more details or have any questions? 

Here are some additional questions that relate to these concepts:
1. What is the general form of a parabola opening horizontally?
2. How do you find the axis of symmetry for any parabola?
3. What is the difference between the focus and the directrix in terms of their roles?
4. How does the value of \(p\) influence the width of the parabola?
5. How would the equation change if the parabola was translated?

**Tip:** Always check the sign of the coefficient of the squared term to determine the direction in which a parabola opens!

Which way does the parabola open? (x - 3)^2 = -4(y + 3). Find the vertex, focus, and directrix for the parabola x^2 = 24y.

Learn how to determine the direction of a parabola opening and how to find the vertex, focus, and directrix for parabolas. This problem is suitable for high school students studying conic sections.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation