Question

Use the table to answer the question. x y −5 8 −7 14 Determine the rate of change and the initial value of the linear function given here as a table of values. (1 point) Responses m=−7, b=−3 m equals negative 7 comma b equals negative 3 m=−3, b=−7 m equals negative 3 comma b equals negative 7 m=−0.5, b=5.5 m equals negative 0.5 comma b equals 5.5 m=−0.5, b=10.5

Answer 1

To find the rate of change (slope, \(m\)) and the initial value (y-intercept, \(b\)) of a linear function from a table of values, we can use the values given in the table:

- For \(x = -5\), \(y = 8\)
- For \(x = -7\), \(y = 14\)

First, we calculate the rate of change (slope, \(m\)):
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points \((-5, 8)\) and \((-7, 14)\):
- \(y_1 = 8\), \(x_1 = -5\)
- \(y_2 = 14\), \(x_2 = -7\)

Substituting in the values:
\[
m = \frac{14 - 8}{-7 - (-5)} = \frac{6}{-2} = -3
\]

Next, we can use the slope and one of the points to find the y-intercept (\(b\)) using the equation of a line in slope-intercept form, \(y = mx + b\). We can use the point \((-5, 8)\):
\[
8 = -3(-5) + b
\]
\[
8 = 15 + b
\]
\[
b = 8 - 15 = -7
\]

Thus, \(m = -3\) and \(b = -7\). Therefore, the rate of change and the initial value of the function are:
\[
m = -3, b = -7
\]

The correct response is:
**m = -3, b = -7**.