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what is the domain and range of this graph 

The graph shown appears to be a parabola that opens upwards. Based on the given information, it has its vertex at the point \((-1, -4)\). Let's determine the domain and range of the function represented by this graph.

### Domain

The domain of a function is the set of all possible \( x \)-values (inputs) for which the function is defined. For a parabola that extends infinitely to the left and right (which is the case here), the domain is all real numbers. In interval notation, this is represented as:

\[
\text{Domain: } (-\infty, \infty)
\]

### Range

The range of a function is the set of all possible \( y \)-values (outputs). For this upward-opening parabola, the lowest point on the graph is the vertex at \((-1, -4)\). Since the parabola opens upwards, the \( y \)-values start from \(-4\) and increase to positive infinity. In interval notation, the range is:

\[
\text{Range: } [-4, \infty)
\]

### Summary

- **Domain:** \((-\infty, \infty)\)
- **Range:** \([-4, \infty)\)

These domain and range values make sense because the parabola continues indefinitely in both the left and right directions for the domain and upwards for the range.

Would you like more details or have any questions about this process?

Here are 5 related questions for further exploration:

1. How would the domain and range change if the parabola opened downwards instead?
2. What is the vertex form of a quadratic equation, and how can it help identify the vertex of a parabola?
3. How would the domain and range be affected if the parabola was shifted vertically or horizontally?
4. Can a parabola have a limited domain and range? If so, under what conditions?
5. How can you find the axis of symmetry for a given parabola?

**Tip:** The vertex of a parabola in standard form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). This is useful for determining the vertex when the quadratic function is not in vertex form.

Explore the domain and range of a parabola with this detailed explanation. Learn how to identify the domain as all real numbers and the range as starting from -4 and extending to infinity. Suitable for high school students.

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