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.