Question

Check if you are correct by writing a function that models the data in terms of "x"
. the graph shows a line with a point of (0, 500) and a point of (1, 450) and a point at (2, 405)

Answer 1

To model the data you provided, we can assume that the points you specified fall on a straight line. The points are (0, 500), (1, 450), and (2, 405). We can use linear regression to find the best-fit line or simply derive the slope and y-intercept from the given points.

### Deriving the Equation of the Line

1. **Determine the slope (\(m\))**:
The slope formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Let's use the points \((0, 500)\) and \((1, 450)\):
\[
m = \frac{450 - 500}{1 - 0} = \frac{-50}{1} = -50
\]

Now, let's verify it with the points \((1, 450)\) and \((2, 405)\):
\[
m = \frac{405 - 450}{2 - 1} = \frac{-45}{1} = -45
\]

Since the slopes between the points are different, the data likely does not follow a simple linear trend, so we will calculate a linear equation based on an approximation.

2. **Let’s use two points for a linear model**.

Using \((0, 500)\) and \((1, 450)\), since they are close, we will use these for our linear approximation.

3. **Find the equation of the line**:
We can use the point-slope form of the linear equation:
\[
y - y_1 = m(x - x_1)
\]

Using point \((0, 500)\) with slope \(m = -50\):
\[
y - 500 = -50(x - 0)
\]
\[
y = -50x + 500
\]

### Implementing in Python

Now, let's implement this as a function in Python.

```python
def model_data(x):
return -50 * x + 500

# Test the function with the given points
test_points = [0, 1, 2]
outputs = {x: model_data(x) for x in test_points}

outputs
```

### Testing the function

When you run this code, it will compute the output for \(x = 0\), \(x = 1\), and \(x = 2\) as follows:

- For \(x = 0\): \( y = -50(0) + 500 = 500 \)
- For \(x = 1\): \( y = -50(1) + 500 = 450 \)
- For \(x = 2\): \( y = -50(2) + 500 = 400 \)

The output dictionary will give you the corresponding values:

```python
{
0: 500,
1: 450,
2: 400
}
```

This function models the data based on a linear approximation through the first two points as discussed. The point at (2, 405) indicates that the linear model could be improved with a non-linear regression, but this gives a reasonable approximation based on the data provided.