To find the rate of change (\(m\)) and the initial value (\(b\)) based on the given values from the table, we can follow these steps:
From the given points:
- When \(x = 1\), \(y = 4\) (point 1: (1, 4))
- When \(x = 3\), \(y = 8\) (point 2: (3, 8))
Step 1: Calculate the rate of change (\(m\))
The formula for the rate of change (slope) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the coordinates of the points:
- \(y_1 = 4\), \(y_2 = 8\)
- \(x_1 = 1\), \(x_2 = 3\)
Substituting the values into the formula:
\[ m = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2 \]
So, the rate of change is \(m = 2\).
Step 2: Determine the initial value (\(b\))
The initial value or y-intercept (\(b\)) in a linear function can be found using the slope-intercept form of the equation, \(y = mx + b\).
Using one of the points to solve for \(b\). Let's use the point (1, 4):
\[ 4 = 2(1) + b \] \[ 4 = 2 + b \] \[ b = 4 - 2 = 2 \]
So, the initial value is \(b = 2\).
Final Answer:
- The rate of change is \(m = 2\), and the initial value is \(b = 2\).