Thank you for sharing the graph and table. I will go through the details of each function $f(x)$ and $g(x)$ based on the provided information. Here’s what we can infer from each component:

Observations about the Graph of $f(x)$:

1. The function $f(x)$ shown in the graph appears to be a quadratic function due to its parabolic shape.
2. The parabola opens upward, indicating that the leading coefficient of the quadratic function is positive.
3. The vertex of $f(x)$ seems to be located at approximately $(-1, -4)$, which suggests that the minimum value of $f(x)$ occurs at $x = -1$.
4. The axis of symmetry for the graph is likely $x = -1$.
5. The function appears symmetric about this axis, as expected for a quadratic function.

Observations about the Table for $g(x)$:

The table provides values of $g(x)$ for specific $x$-values. We can analyze these values to understand the function $g(x)$.

1. The values in the table suggest that $g(x)$ also represents a quadratic function because the outputs (values of $g(x)$) follow a parabolic trend.
2. The smallest value of $g(x)$ occurs at $x = 1$, with $g(1) = 3$. This indicates that the vertex of $g(x)$ is likely at $x = 1$.
3. The values are symmetric around $x = 1$, which is a common feature of quadratic functions centered at their vertex.
4. Based on the table, $g(x)$ also appears to be an upward-opening parabola, similar to $f(x)$.

Questions

Based on the graph and table, please answer the following questions:

1. What is the minimum value of $f(x)$ according to the graph?
2. What is the vertex of the function $g(x)$ based on the table?
3. Do $f(x)$ and $g(x)$ have the same vertex?
4. At which points do $f(x)$ and $g(x)$ share the same $x$-values for corresponding $y$-values?
5. Is the symmetry of $f(x)$ and $g(x)$ about their respective vertices consistent?

Tip:

When analyzing quadratic functions, identifying the vertex is crucial, as it reveals the function's maximum or minimum point and helps understand the symmetry of the function.