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The equation given is \( y = x^2 + 4 \), which is a quadratic equation of the form \( y = ax^2 + c \) where \( a = 1 \) and \( c = 4 \).

Here’s how we can analyze this to identify the correct graph:

1. **Shape of the graph**: Since \( a = 1 \), the graph will be a **parabola that opens upwards** (as \( a > 0 \)).

2. **Vertex**: The vertex form of a quadratic function \( y = (x - h)^2 + k \) shows that the vertex of the parabola is at \( (h, k) \). Here, \( h = 0 \) and \( k = 4 \), meaning the vertex of the parabola is at \( (0, 4) \).

3. **Y-intercept**: The graph intersects the y-axis at \( (0, 4) \).

4. **General behavior**: Since this is a simple upward-facing parabola with its vertex shifted 4 units up from the origin, the graph will rise symmetrically on both sides.

Now, let’s examine the options:
- Option **b)** correctly shows a parabola opening upwards with its vertex at \( (0, 4) \), which corresponds to \( y = x^2 + 4 \).
- Other options either open downwards or have the wrong vertex location.

Thus, **option b)** is the correct graph for \( y = x^2 + 4 \).

Would you like more details or have any questions?

Here are some follow-up questions:
1. What happens to the graph of \( y = x^2 + c \) as \( c \) increases or decreases?
2. How does changing the coefficient \( a \) affect the width of the parabola?
3. What is the equation of a parabola that opens downwards with vertex at \( (0, -4) \)?
4. How would you graph \( y = (x - 2)^2 + 4 \)?
5. What are the roots of \( y = x^2 + 4 \)?

**Tip**: The value of \( c \) in the equation \( y = ax^2 + c \) shifts the parabola vertically by \( c \) units.

Which one of the following could be the graph of y = x^2 + 4?

Explore how to identify the correct graph for the quadratic equation y = x^2 + 4. This lesson covers algebraic concepts including parabola transformations and understanding vertex positions. Perfect for high school students learning quadratic functions.

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