https://file.aiquickdraw.com/mathbot/file/1725881959337w9g9fkun.jpg

Let's break down the questions in the image:

### 8. Identifying if the table of values represents a quadratic function.

The table of values provides:

\[
\begin{array}{c|c}
x & y \\
\hline
-2 & 5 \\
-1 & 3 \\
0  & 1 \\
1  & 3 \\
2  & 5 \\
3  & 7 \\
\end{array}
\]

To determine if the function is quadratic, we can calculate the first and second differences of the \(y\)-values:

- First differences (differences between consecutive \(y\)-values):
  \[
  3 - 5 = -2,\quad 1 - 3 = -2,\quad 3 - 1 = 2,\quad 5 - 3 = 2,\quad 7 - 5 = 2
  \]

- Second differences (differences between consecutive first differences):
  \[
  -2 - (-2) = 0,\quad 2 - (-2) = 4,\quad 2 - 2 = 0
  \]

Since the second differences are not constant (they go from 0 to 4 to 0), this is not a quadratic function.

### D. Determine if the given graphs are quadratic functions.

#### Graph 9
This graph is a parabola opening upwards. Parabolas represent quadratic functions of the form \(y = ax^2 + bx + c\). Thus, **Graph 9 represents a quadratic function**.

**Domain**: All real numbers, \( (-\infty, \infty) \).
**Range**: Since the parabola opens upwards and has a minimum at the vertex, the range is \( [0, \infty) \).

#### Graph 10
The second graph shows the equation \(y^2 = -4x\). This is not a quadratic function because it represents a sideways parabola, which is actually a relation and not a function in terms of \(x\). **Graph 10 does not represent a quadratic function**.

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Would you like further clarification on any of the above concepts?

### 5 Relative Questions:
1. What defines a quadratic function based on its equation?
2. How can you determine the domain and range of a quadratic function?
3. Can a sideways parabola ever represent a function? Why or why not?
4. Why are second differences used to verify if a sequence represents a quadratic function?
5. How do the properties of a quadratic graph change depending on the sign of the leading coefficient?

### Tip:
Always verify if the graph of a function is quadratic by checking if it forms a parabola that opens either upwards or downwards!

Determine whether each of the graphs represents a quadratic function or not. If it is a quadratic function, identify its domain and range.

Learn how to identify quadratic functions from graphs and how to determine their domain and range. This solution focuses on the parabolas, quadratic functions, and algebraic concepts suitable for students in Grades 9-12.

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