To determine the initial value and rate of change (slope) of the linear function defined by the given points \((-4, -\frac{10}{3})\) and \((-1, -\frac{4}{3})\), let's use the formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Step 1: Substitute the values into the formula
Let \((x_1, y_1) = (-4, -\frac{10}{3})\) and \((x_2, y_2) = (-1, -\frac{4}{3})\).
Continuing with the calculation, we have:
- \(y_2 = -\frac{4}{3}\)
- \(y_1 = -\frac{10}{3}\)
- \(x_2 = -1\)
- \(x_1 = -4\)
Step 2: Calculate the slope (rate of change)
\[ m = \frac{-\frac{4}{3} - \left(-\frac{10}{3}\right)}{-1 - (-4)} \]
This simplifies to:
\[ m = \frac{-\frac{4}{3} + \frac{10}{3}}{-1 + 4} = \frac{\frac{6}{3}}{3} = \frac{2}{3} \]
Step 3: Identify the initial value
To express the linear function in the slope-intercept form \(y = mx + b\), we need to find the y-intercept \(b\) (initial value).
We can use one of the points and the slope to solve for \(b\). Using the point \((-4, -\frac{10}{3})\):
\[ y = mx + b \] \[ -\frac{10}{3} = \frac{2}{3}(-4) + b \]
Calculating \( \frac{2}{3}(-4) \):
\[ \frac{2}{3} \cdot -4 = -\frac{8}{3} \]
So the equation becomes:
\[ -\frac{10}{3} = -\frac{8}{3} + b \]
Adding \(\frac{8}{3}\) to both sides:
\[ b = -\frac{10}{3} + \frac{8}{3} = -\frac{2}{3} \]
Summary of Results
- Rate of Change (Slope): \( \frac{2}{3} \)
- Initial Value (y-intercept): \( -\frac{2}{3} \)
Thus, the linear function can be expressed as:
\[ y = \frac{2}{3}x - \frac{2}{3} \]