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Determine the domain on which the following function is increasing.


To determine the domain on which the given function is increasing, we first need to analyze the graph of the function provided in the image.

### Analysis:
1. **Shape of the Graph**: The graph appears to be a parabola that opens downwards, indicating that it is a quadratic function with a negative leading coefficient.

2. **Vertex of the Parabola**: The vertex is the highest point on the parabola. The function increases on the interval leading up to the vertex and decreases afterwards.

3. **Increasing Interval**: The function is increasing on the interval where the graph moves upwards as you move from left to right.

### Solution:
- **Find the vertex**: Based on the graph, the vertex appears to be at \( x = 0 \). 
- **Determine the interval of increase**: The function increases from the leftmost point on the x-axis up to the vertex.

Thus, the function is increasing on the interval:
\[
(-\infty, 0]
\]

Would you like more details or have any other questions?

### Related Questions:
1. How do you find the vertex of a parabola algebraically?
2. What does it mean for a function to be increasing on an interval?
3. Can you explain how to determine the domain of a function from its graph?
4. How does the coefficient of the quadratic term affect the direction the parabola opens?
5. What are the implications of the vertex being at \( x = 0 \)?

### Tip:
For quadratic functions, the vertex can be found using the formula \( x = -\frac{b}{2a} \) when the function is in the form \( ax^2 + bx + c \). This helps in determining the intervals of increase or decrease easily.

Determine the domain on which the following function is increasing.

This problem involves determining the interval on which a quadratic function is increasing, based on its graph. It covers key algebraic concepts including the vertex of a parabola, the properties of quadratic functions, and how to identify intervals of increase and decrease. Suitable for students in Grades 9-10.

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