Question

Using the table below, determine the Rate of Change and the Initial Value

x y
0 50
1 250
2 450
3 650

Answer 1

To determine the Rate of Change and the Initial Value from the given table, we can analyze the relationship between the variables \( x \) and \( y \).

### Initial Value
The Initial Value is the value of \( y \) when \( x = 0 \). From the table:

\[
\text{Initial Value} = y \text{ when } x = 0 = 50
\]

### Rate of Change
The Rate of Change can be calculated by determining how much \( y \) changes for each unit increase in \( x \). We can find the rate of change using the formula:

\[
\text{Rate of Change} = \frac{\text{Change in } y}{\text{Change in } x}
\]

We can calculate this between any two points. Let's use the first two points (for simplicity):

- At \( x = 0 \), \( y = 50 \)
- At \( x = 1 \), \( y = 250 \)

Calculating the change:

\[
\text{Change in } y = 250 - 50 = 200
\]
\[
\text{Change in } x = 1 - 0 = 1
\]
\[
\text{Rate of Change} = \frac{200}{1} = 200
\]

We can verify this for other pairs of points:

1. From \( x = 1 \) to \( x = 2 \):
- \( y \) changes from \( 250 \) to \( 450 \), hence:
\[
\text{Change in } y = 450 - 250 = 200, \quad \text{Change in } x = 2 - 1 = 1, \quad \text{Rate of Change} = \frac{200}{1} = 200
\]

2. From \( x = 2 \) to \( x = 3 \):
- \( y \) changes from \( 450 \) to \( 650 \), hence:
\[
\text{Change in } y = 650 - 450 = 200, \quad \text{Change in } x = 3 - 2 = 1, \quad \text{Rate of Change} = \frac{200}{1} = 200
\]

### Conclusion
The Rate of Change is consistent, thus:

- **Initial Value:** \( 50 \)
- **Rate of Change:** \( 200 \)