A table showing pairs of  x and  y values is shown below.

x	y
0	0.5
1	1.5
2	4.5
3	13.5
4	40.5
Which equation describes the relationship between the pairs of x and y values?
A. y = (0.5-3)
B. y = 0.5.3
C. y=2x0.5
D. y = x +0.5

Let's carefully analyze the given values and identify the relationship between \(x\) and \(y\). 

**Table of values:**
\[
\begin{aligned}
x & : 0, 1, 2, 3, 4 \\
y & : 0.5, 1.5, 4.5, 13.5, 40.5
\end{aligned}
\]

### Identifying the pattern:

- Notice how the \(y\)-values seem to increase rapidly, hinting at a potential **exponential** or **multiplicative relationship**.  
- Let's examine the ratio between consecutive \(y\)-values:

  \[
  \frac{1.5}{0.5} = 3, \quad \frac{4.5}{1.5} = 3, \quad \frac{13.5}{4.5} = 3, \quad \frac{40.5}{13.5} = 3
  \]

- The \(y\)-values are multiplied by **3** each time \(x\) increases by 1. 

This suggests the relationship follows the **exponential pattern**:
\[
y = 0.5 \cdot 3^x
\]

### Checking the answer options:
- **A. \(y = (0.5 - 3)\)**: Incorrect, this is not a valid equation.
- **B. \(y = 0.5 \cdot 3\)**: Incorrect, it is a constant value rather than a function of \(x\).
- **C. \(y = 2x0.5\)**: Incorrect, it’s not exponential and doesn't fit the data.
- **D. \(y = 0.5 \cdot 3^x\)**: This equation matches our findings perfectly!

Thus, the correct answer is:

\[
\boxed{D. \, y = 0.5 \cdot 3^x}
\]

---

Would you like more details or have any questions?  

Here are 5 related questions to deepen your understanding:
1. What is the general form of an exponential function?
2. How do you determine if a relationship is linear or exponential from a table of values?
3. What is the difference between an arithmetic and a geometric sequence?
4. How would the equation change if the initial \(y\)-value were 1 instead of 0.5?
5. Can exponential functions have negative values for \(y\)? Why or why not?

**Tip:** When the ratio between consecutive outputs is constant, the function is likely exponential.

A table showing pairs of x and y values is shown below.

x	y
0	0.5
1	1.5
2	4.5
3	13.5
4	40.5
Which equation describes the relationship between the pairs of x and y values?
A. y = (0.5-3)
B. y = 0.5.3
C. y = 2x0.5
D. y = x +0.5

Explore how to find the equation that describes the exponential relationship between x and y value pairs, using the function y = 0.5 * 3^x. This problem is designed for high school students and covers key algebraic concepts and exponential growth patterns.

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