To determine the domain on which the function is increasing, let's analyze the graph provided.

The graph appears to be a downward-facing parabola, which suggests it is a quadratic function. Quadratic functions are typically increasing up to their vertex and then decreasing thereafter (for downward-facing parabolas).

Step-by-Step Analysis:

1. Identify the Vertex: The vertex of this parabola is at the point $(1, 4)$, where $x = 1$ and $y = 4$.
2. Behavior of the Parabola:
   • From $x = -\infty$ to $x = 1$, the function is increasing, as the values of $y$ rise as $x$ approaches 1.
   • After $x = 1$, the function begins to decrease, as the parabola opens downward.

Conclusion:

The function is increasing on the interval: $(-\infty, 1]$ This interval represents the domain where the function is increasing.

Would you like more details or have any questions?

Here are five related questions for further exploration:

1. How do we determine the interval of increase and decrease for other types of functions?
2. How can we identify the vertex of a parabola algebraically?
3. What is the significance of the vertex in the context of a quadratic function's graph?
4. How does the direction of a parabola (upward vs. downward) affect its increasing and decreasing intervals?
5. How can we find intervals of increase and decrease for functions other than quadratics?

Tip: When analyzing functions graphically, remember that an upward trend on a graph indicates an increasing interval, while a downward trend indicates a decreasing interval.