(-3,4), (0,3), (3,2), (6,1) what’s the initial value and the rate of change

Answer 1

To determine the initial value and the rate of change for the given points \((-3,4)\), \((0,3)\), \((3,2)\), and \((6,1)\), you can treat these points as part of a linear function of the form \(y = mx + b\), where \(m\) is the rate of change (slope) and \(b\) is the initial value (y-intercept).

### Rate of Change (Slope)
The slope \(m\) is defined as the change in \(y\) divided by the change in \(x\) between any two points:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the points \((0,3)\) and \((3,2)\):
\[ m = \frac{2 - 3}{3 - 0} = \frac{-1}{3} = -\frac{1}{3} \]

You can verify this slope with other pairs of points to ensure consistency:

Using the points \((3,2)\) and \((6,1)\):
\[ m = \frac{1 - 2}{6 - 3} = \frac{-1}{3} = -\frac{1}{3} \]

Using the points \((-3,4)\) and \((0,3)\):
\[ m = \frac{3 - 4}{0 - (-3)} = \frac{-1}{3} = -\frac{1}{3} \]

The slope is consistently \(-\frac{1}{3}\).

### Initial Value (Y-intercept)
The y-intercept \(b\) is the value of \(y\) when \(x = 0\). From the given point \((0,3)\):
\[ b = 3 \]

### Equation of the Line
Now we can form the equation of the line using the slope \(m = -\frac{1}{3}\) and y-intercept \(b = 3\):
\[ y = -\frac{1}{3}x + 3 \]

### Summary
- **Initial Value (Y-intercept):** \(3\)
- **Rate of Change (Slope):** \(-\frac{1}{3}\)

The linear function representing these points is \(y = -\frac{1}{3}x + 3\).