Problem 1

Explanation

We will analyze each statement based on the given graph.

Analysis of Statement 1

Based on the graph, it is clear that the curve intersects the $x$-axis at the point $(4, 0)$.

Thus, statement $(1)$ is correct.

Analysis of Statement 2

Based on the graph, the curve intersects the $y$-axis at two points, namely at $y = 3$ and $y = -5$. So the intersection points are $(0, 3)$ and $(0, -5)$.

Statement $(2)$ states that the intersection points are $(0, -5)$ and $(0, 1)$. Since $(0, 3) \neq (0, 1)$, this statement does not match the graph.

Thus, statement $(2)$ is incorrect.

Analysis of Statement 3

The axis of symmetry for a horizontal parabola can be determined from the midpoint between the two $y$-intercepts (if known).

The axis of symmetry is obtained as the line $y = -1$.

Thus, statement $(3)$ is correct.

Analysis of Statement 4

Since the axis of symmetry is the horizontal line $y = -1$, the statement that the axis of symmetry is $x = -3$ (a vertical line) is clearly wrong.

Thus, statement $(4)$ is incorrect.

Conclusion

The correct statements are statements $(1)$ and $(3)$.