https://file.aiquickdraw.com/mathbot/file/1729016894877i19kw8q1.jpg

In the image, you are asked to identify which graph represents the given table of values for \(x\) and \(y\). The table of values is:

\[
\begin{array}{c|c}
x & y \\
\hline
-3 & 15 \\
-2 & 9 \\
-1 & 3 \\
0 & -1 \\
1 & 3 \\
3 & 9 \\
\end{array}
\]

This appears to represent a quadratic relationship (as the \(y\)-values decrease and then increase again symmetrically). Let's analyze the dataIn the image, you are asked to identify which graph represents the given table of values for \(x\) and \(y\). The table of values is:

\[
\begin{array}{c|c}
x & y \\
\hline
-3 & 15 \\
-2 & 9 \\
-1 & 3 \\
0 & -1 \\
1 & 3 \\
3 & 9 \\
\end{array}
\]

This appears to represent a quadratic relationship (as the \(y\)-values decrease and then increase again symmetrically). Let's analyze the 

- At \(x = 0\), the \(y\)-value is \(-1\), which likely corresponds to the vertex of the parabola.
- As \(x\) moves away from 0 in both directions (either positive or negative), the \(y\)-values increase, which indicates the parabola is opening upwards.

Next, let's calculate a few points:

1. The symmetry about \(x = 0\) suggests the graph is centered there, typical of a parabola.
2. The values of \(y\) increase symmetrically as \(x\) moves away from the vertex.

From this, we can conclude that the correct graph should represent a parabola opening upwards, with the vertex at \((0, -1)\) and symmetric points at \(x = 1, -1\) with \(y = 3\), and at \(x = 2, -2\) with \(y = 9\).

By matching this description with the graphs in the image, you should look for the one that has these characteristics. 

Let me know if you want further details or have any other questions.

Here are five related questions to expand your understanding:

1. How can we derive the equation of the parabola from the given points?
2. What role does the vertex form of a parabola play in identifying the graph?
3. How would the graph change if the vertex shifted to another point?
4. What kind of symmetry do all parabolas exhibit, and how can it be identified?
5. How can you determine whether a parabola opens upwards or downwards just by looking at a table of values?

**Tip:** When identifying a graph, always check the vertex and symmetry to narrow down the options.

Which graph best represents the table of x and y values below?

Learn how to identify the correct graph for a quadratic function based on a table of x and y values. This problem demonstrates how to analyze the vertex and symmetry of a parabola for students in Grade 9.

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