To determine the rate of change (m) and the initial value (b) of the linear function given the table of values:
We have the points:
- (-5, 8)
- (-7, 14)
The rate of change (slope) \( m \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (-5, 8) and (-7, 14):
- \( (x_1, y_1) = (-5, 8) \)
- \( (x_2, y_2) = (-7, 14) \)
Substituting the values into the formula:
\[ m = \frac{14 - 8}{-7 - (-5)} = \frac{6}{-2} = -3 \]
Now we can find the initial value \( b \) using the slope-intercept form of a line \( y = mx + b \). We can substitute one of the points into this equation to solve for \( b \). Let's use the point (-5, 8):
\[ 8 = -3(-5) + b \]
Calculating this:
\[ 8 = 15 + b \] \[ b = 8 - 15 = -7 \]
Thus, the rate of change (m) is -3 and the initial value (b) is -7. Therefore, the answer is:
m=−3, b=−7