To solve for \( y \) in the given quadratic functions, we'll find the key features of each quadratic, such as the vertex, axis of symmetry, and direction in which the parabola opens. This helps us understand the graph's behavior.

### 1. Function: \( y = (x + 2)^2 - 3 \)

- **Vertex form**: The function is already in vertex form: \( y = a(x - h)^2 + k \), where \( a = 1 \), \( h = -2 \), and \( k = -3 \).
- **Vertex**: \( (-2, -3) \)
- **Axis of symmetry**: \( x = -2 \)
- **Direction**: Opens upwards since \( a = 1 > 0 \).

To graph this function:
1. Start at the vertex \((-2, -3)\).
2. Since \( a = 1 \), the parabola opens upwards.
3. Plot additional points by choosing x-values around the vertex:
   - If \( x = -1 \), \( y = (-1 + 2)^2 - 3 = 1 - 3 = -2 \).
   - If \( x = 0 \), \( y = (0 + 2)^2 - 3 = 4 - 3 = 1 \).
4. Reflect these points across the axis of symmetry \( x = -2 \).

### 2. Function: \( y = -(x - 5)^2 + 3 \)

- **Vertex form**: The function is in vertex form: \( y = a(x - h)^2 + k \), where \( a = -1 \), \( h = 5 \), and \( k = 3 \).
- **Vertex**: \( (5, 3) \)
- **Axis of symmetry**: \( x = 5 \)
- **Direction**: Opens downwards since \( a = -1 < 0 \).

To graph this function:
1. Start at the vertex \((5, 3)\).
2. Since \( a = -1 \), the parabola opens downwards.
3. Plot additional points by choosing x-values around the vertex:
   - If \( x = 4 \), \( y = -(4 - 5)^2 + 3 = -1 + 3 = 2 \).
   - If \( x = 6 \), \( y = -(6 - 5)^2 + 3 = -1 + 3 = 2 \).
4. Reflect these points across the axis of symmetry \( x = 5 \).

### 3. Function: \( y = (x - 4)^2 - 5 \)

- **Vertex form**: The function is in vertex form: \( y = a(x - h)^2 + k \), where \( a = 1 \), \( h = 4 \), and \( k = -5 \).
- **Vertex**: \( (4, -5) \)
- **Axis of symmetry**: \( x = 4 \)
- **Direction**: Opens upwards since \( a = 1 > 0 \).

To graph this function:
1. Start at the vertex \((4, -5)\).
2. Since \( a = 1 \), the parabola opens upwards.
3. Plot additional points by choosing x-values around the vertex:
   - If \( x = 3 \), \( y = (3 - 4)^2 - 5 = 1 - 5 = -4 \).
   - If \( x = 5 \), \( y = (5 - 4)^2 - 5 = 1 - 5 = -4 \).
4. Reflect these points across the axis of symmetry \( x = 4 \).

### Summary

By identifying the vertex and plotting a few points around it, you can easily sketch the graph of each quadratic function. Ensure to plot the vertex first and then use symmetry and additional points to define the shape.

Would you like further details or have any other questions about graphing these quadratic functions?

### Further Questions

1. What is the significance of the direction in which a parabola opens when analyzing a quadratic function?
2. How does changing the \( k \) value in the vertex form of a quadratic function affect its graph?
3. Can a quadratic function have more than one vertex? Why or why not?
4. How does the axis of symmetry help in graphing a quadratic function?
5. What are the steps to convert a quadratic function from standard form to vertex form?

**Tip:** When sketching quadratic graphs, always start by identifying the vertex and plotting points symmetrically around it for accurate representation.

Learn how to solve for y in graphing quadratic functions using vertex form analysis. Explore examples such as y = (x + 2)^2 - 3, y = -(x - 5)^2 + 3, and y = (x - 4)^2 - 5 with step-by-step solutions and graphing techniques. Ideal for high school students and anyone studying quadratic functions.

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