Use the table to answer the question.

Question

x
x
y
y
−5
−
5
8
−7
−
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=−3, b=−7
m
=
−
3
,

b
=
−
7
m equals negative 3 comma b equals negative 7

m=−0.5, b=5.5
m
=
−
0.5
,

b
=
5.5
m equals negative 0.5 comma b equals 5.5

m=−7, b=−3
m
=
−
7
,

b
=
−
3
m equals negative 7 comma b equals negative 3

m=−0.5, b=10.5

Answer 1

To determine the rate of change (slope, \(m\)) and the initial value (y-intercept, \(b\)) of the linear function, we can use the values provided in the table.

Based on the provided coordinates:

• The first point is \((-5, 8)\)
• The second point is \((-7, 14)\)

1. Calculate the slope \(m\) using the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Plugging in the values from the points \((-5, 8)\) and \((-7, 14)\): \[ m = \frac{14 - 8}{-7 - (-5)} = \frac{6}{-2} = -3 \]

2. Determine the y-intercept \(b\). We can use one of the points and the slope to find \(b\). Using the point \((-5, 8)\): \[ y = mx + b \] Where \(y = 8\), \(m = -3\), and \(x = -5\): \[ 8 = -3(-5) + b \] \[ 8 = 15 + b \] \[ b = 8 - 15 = -7 \]

Thus, the rate of change (slope) is \(m = -3\) and the initial value (y-intercept) is \(b = -7\).

The answer is: \(m = -3, b = -7\).