The equation given is:

\[
(x + 5)^2 = y
\]

This represents a parabola in the Cartesian coordinate system. To understand the properties of this equation, let's analyze it step-by-step.

### 1. Expanded Form

We can expand the squared term to get the standard form of the equation:

\[
(x + 5)^2 = y \implies x^2 + 10x + 25 = y
\]

### 2. Vertex Form

The original equation \((x + 5)^2 = y\) is already in vertex form:

\[
y = (x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola. By comparing:

- \(h = -5\)
- \(k = 0\)

Thus, the vertex of this parabola is at \((-5, 0)\).

### 3. Direction of the Parabola

Since the \(x\) term is squared and there are no modifications to the \(y\) term other than being set equal to the squared term, the parabola opens **upwards**.

### 4. Axis of Symmetry

The axis of symmetry for a parabola in the form \(y = (x + 5)^2\) is given by:

\[
x = -5
\]

### 5. Key Features of the Parabola

- **Vertex**: \((-5, 0)\)
- **Axis of Symmetry**: \(x = -5\)
- **Direction**: Upward-opening
- **Intercepts**: 
  - **y-intercept**: When \(x = 0\),
  
  \[
  y = (0 + 5)^2 = 25 \implies (0, 25)
  \]

  - **x-intercepts**: Solving for \(x\) when \(y = 0\), 

  \[
  (x + 5)^2 = 0 \implies x + 5 = 0 \implies x = -5
  \]

  So, the x-intercept is at \((-5, 0)\), which is the vertex.

Would you like more details or have any questions? Here are some related questions that might interest you:

1. How do you find the focus and directrix of a parabola?
2. What are the properties of parabolas and how do they relate to their equations?
3. How do you convert from standard form to vertex form of a parabola?
4. What is the general formula for a parabola that opens sideways?
5. How can parabolas be used to model real-world situations?

**Tip:** Remember that the vertex form of a parabola makes it easy to identify the vertex and the direction in which the parabola opens.

Explore the equation (x + 5)^2 = y and understand its properties, vertex, symmetry, and more with a detailed explanation. This problem covers key algebraic concepts, including quadratic equations and vertex form. Suitable for high school students.

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