Question

Determine the initial value and the rate of change of the linear function as given in the graph. The slope is positive, and the poiints are (-1.5,3), (0,0), and (0.5,1)

Answer 1

To determine the initial value and the rate of change (slope) of the linear function given the points \((-1.5, 3)\), \((0, 0)\), and \((0.5, 1)\), we will start by identifying the greatest insights that can be gathered from these points.

1. **Initial Value**: The initial value of a linear function is typically the value of the function when \(x = 0\). From the points provided:

- At \((0, 0)\), the function has a value of \(0\).

Therefore, the initial value (the y-intercept) is \(0\).

2. **Rate of Change (Slope)**: The slope of a linear function can be calculated using any two points on the line. The formula for calculating the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

We can use the points \((0, 0)\) and \((0.5, 1)\):

Let \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (0.5, 1)\):

\[
m = \frac{1 - 0}{0.5 - 0} = \frac{1}{0.5} = 2
\]

We could also check the slope using the points \((0, 0)\) and \((-1.5, 3)\):

Let \((x_1, y_1) = (0, 0)\) and \((x_2, y_2) = (-1.5, 3)\):

\[
m = \frac{3 - 0}{-1.5 - 0} = \frac{3}{-1.5} = -2
\]

Clearly, this calculation gives a negative slope, which indicates that something may be amiss with the connection between the points. Notably, points must be organized to form a linear relationship, suggesting potentially misplaced values.

3. **We can consider just the pairs with unambiguous linear approach**: Let's analyze pairs of points:

Using \((0, 0)\) and \((0.5, 1)\), we reaffirmed the slope as \(2\). As a convenience and conciseness, it remains a sound basis for any slope interpretation:

Thus, the rate of change of the function is \(2\), and the function can be expressed as:

\[
y = 2x
\]

In conclusion:

- **Initial Value**: \(0\)
- **Rate of Change**: \(2\)