https://file.aiquickdraw.com/mathbot/file/1723677904118dejvqk39.jpg

Is this linear, quadratic or exponential??

To determine whether the data in the table represents a linear, quadratic, or exponential relationship, we can examine the pattern in the \(y\) values as \(x\) changes. 

Here’s the table:

\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
-2 & 1.5 \\
-1 & 3.5 \\
0 & 5.5 \\
1 & 7.5 \\
2 & 9.5 \\
\hline
\end{array}
\]

### Step 1: Check for a Linear Relationship
A linear relationship follows the equation \(y = mx + b\), where \(m\) is the slope. For a linear function, the difference in \(y\) values (the first differences) should be constant as \(x\) changes.

Let's calculate the first differences:

\[
\text{First Differences:} \quad \Delta y = y_{i+1} - y_i
\]

- From \(x = -2\) to \(x = -1\): \(3.5 - 1.5 = 2.0\)
- From \(x = -1\) to \(x = 0\): \(5.5 - 3.5 = 2.0\)
- From \(x = 0\) to \(x = 1\): \(7.5 - 5.5 = 2.0\)
- From \(x = 1\) to \(x = 2\): \(9.5 - 7.5 = 2.0\)

Since the first differences are constant, the relationship is **linear**.

### Conclusion:
The data in the table represents a linear relationship. The equation of the line can be written as \(y = 2x + 5.5\).

Would you like a more detailed explanation or have any questions about this?

Here are some related questions:
1. How do you find the equation of a line given two points?
2. What does the slope of a line represent in a linear equation?
3. How can you determine if a set of data points are linear by plotting them?
4. What are first differences, and how do they help identify a linear relationship?
5. How do you calculate the y-intercept of a line?
6. Can a quadratic function ever produce constant first differences?
7. What is the difference between linear, quadratic, and exponential relationships?
8. How do you convert a table of data points into a graph?

**Tip:** When checking if data is linear, always calculate the first differences to see if they are constant. This method is a quick way to identify linear relationships.

Explore how to determine if data represents a linear, quadratic, or exponential relationship using first differences. Step-by-step analysis with example data table (-2, 1.5), (-1, 3.5), (0, 5.5), (1, 7.5), (2, 9.5).

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