The graph depicts a parabolic curve that opens upwards, and the vertex is at $(0, -1)$. Here's how we can analyze it:

Part (a): Intervals where the function is increasing

The function is increasing on intervals where the graph slopes upward as $x$ increases. Observing the graph, the function starts increasing from the vertex ($x = 0$) onwards to the right. Hence, the interval is: $(0, \infty)$

Part (b): Intervals where the function is decreasing

The function is decreasing on intervals where the graph slopes downward as $x$ increases. This occurs from the left end of the graph up to the vertex ($x = 0$). Thus, the interval is: $(-\infty, 0)$

Final Answer:

1. Increasing interval: $(0, \infty)$
2. Decreasing interval: $(-\infty, 0)$

Would you like a detailed breakdown of how to write this in interval notation or more explanation on the graph's behavior?

Related Questions:

1. What is the vertex of this parabola and how is it calculated?
2. How can we determine if a function is increasing or decreasing algebraically?
3. What is the domain and range of this function based on the graph?
4. How can symmetry in a graph help identify intervals of increase and decrease?
5. How does this graph represent a quadratic function?

Tip:

For any quadratic function $f(x) = ax^2 + bx + c$, the function decreases until the vertex and increases thereafter if $a > 0$ (upward parabola).