To find the axis of symmetry and analyze the graph you've uploaded, let's break it down:

Observations:

1. Parabola Shape: The curve in the image is a parabola that opens downwards, implying a quadratic function.
2. Vertex: The vertex (top point) of the parabola is located at around $(0, 200)$, as it appears to reach a maximum there.
3. Symmetry: Since the parabola is symmetric and the vertex lies on the y-axis, the axis of symmetry is a vertical line passing through the vertex.
4. Zeroes: The parabola intersects the x-axis at $(0, 0)$, indicating that the roots of the quadratic function are at this point.

Axis of Symmetry:

The axis of symmetry of a parabola can be found using the equation of the vertex form of a parabola:

$x = -\frac{b}{2a}$

However, in this case, we can observe directly from the graph that the vertex is located at $x = 0$. This suggests that the axis of symmetry is:

$x = 0$

Explanation Using the Axis of Symmetry:

• Since the parabola is symmetric around the axis $x = 0$, any point on the graph to the left of $x = 0$ has a corresponding point on the right side at the same vertical level. This symmetry property ensures that for each value of $y$, except at the vertex, the same value appears at $x = -c$ and $x = c$, where $c$ is some distance from the axis of symmetry.
• In this graph, the parabola intersects the x-axis at $(0, 0)$, which is both the vertex and a root of the equation. The vertical axis of symmetry helps to understand how the graph is mirrored on both sides of the y-axis.

Conclusion:

The axis of symmetry for this parabola is the line $x = 0$, which divides the graph into two mirror-image halves. This axis of symmetry is crucial in identifying the symmetry and properties of the quadratic function.

Would you like more details on any part of this analysis?

Here are some related questions to expand on this topic:

1. How do you derive the equation of a parabola from the graph?
2. How does the vertex relate to the maximum or minimum of a parabola?
3. What is the general form of a quadratic function, and how can we use it to find the vertex?
4. What are other properties of parabolas related to their symmetry?
5. How does the direction in which a parabola opens affect its symmetry and properties?

Tip: The axis of symmetry can always be found from the vertex of a parabola, and it plays a key role in understanding quadratic functions.