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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 changes in the \(y\) values as \(x\) changes.

Here is the data from the table:

\[
\begin{array}{c|c}
x & y \\
\hline
-2 & 40 \\
-1 & 20 \\
0 & 10 \\
1 & 5 \\
2 & 2.5 \\
\end{array}
\]

### Step 1: Check for a Linear Relationship
A linear relationship has a constant rate of change, meaning the difference between consecutive \(y\) values should be constant.

Calculate the first differences (change in \(y\) values):
\[
\begin{aligned}
\Delta y_{1} &= 20 - 40 = -20 \\
\Delta y_{2} &= 10 - 20 = -10 \\
\Delta y_{3} &= 5 - 10 = -5 \\
\Delta y_{4} &= 2.5 - 5 = -2.5 \\
\end{aligned}
\]
The first differences are not constant, so the relationship is not linear.

### Step 2: Check for a Quadratic Relationship
A quadratic relationship has a constant second difference.

Calculate the second differences:
\[
\begin{aligned}
\Delta^2 y_{1} &= (-10) - (-20) = 10 \\
\Delta^2 y_{2} &= (-5) - (-10) = 5 \\
\Delta^2 y_{3} &= (-2.5) - (-5) = 2.5 \\
\end{aligned}
\]
The second differences are also not constant, so the relationship is not quadratic.

### Step 3: Check for an Exponential Relationship
An exponential relationship has a constant ratio between consecutive \(y\) values.

Calculate the ratio between consecutive \(y\) values:
\[
\begin{aligned}
\text{Ratio } R_{1} &= \frac{20}{40} = 0.5 \\
\text{Ratio } R_{2} &= \frac{10}{20} = 0.5 \\
\text{Ratio } R_{3} &= \frac{5}{10} = 0.5 \\
\text{Ratio } R_{4} &= \frac{2.5}{5} = 0.5 \\
\end{aligned}
\]
The ratio between consecutive \(y\) values is constant, indicating an **exponential relationship**.

The function appears to follow an exponential decay of the form \(y = a \cdot b^x\) where \(a = 40\) and \(b = 0.5\).

Would you like more details or have any questions?

### Related Questions
1. How can we derive the equation of an exponential function from the data points?
2. What are the characteristics of exponential growth versus exponential decay?
3. How can we verify the type of relationship using a graph?
4. What real-world phenomena can be modeled by exponential decay?
5. How can we convert the given exponential model to its logarithmic form?
6. What is the half-life in the context of exponential decay?
7. Can we model this data using logarithmic transformation?
8. How does the initial value \(a\) influence the exponential function?

### Tip
When identifying the type of function that fits a given data set, checking differences and ratios between consecutive values provides quick insights into whether the relationship is linear, quadratic, or exponential.

Explore how to determine whether a given set of data exhibits a linear, quadratic, or exponential relationship with detailed steps. This example problem uses real data points to illustrate the process of analyzing differences and ratios between consecutive values. Suitable for high school students in Grades 10-12.

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