Use the table to answer the question.

Question

Figuring out the starting value and rate of change from a given value table.
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=−3, b=−7
m equals negative 3 comma b equals negative 7

m=−0.5, b=10.5
m equals negative 0.5 comma b equals 10.5

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

m=−0.5, b=5.5
m equals negative 0.5 comma b equals 5.5
Remaining Attempts : 3
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Answer 1

To determine the rate of change \((m)\) and the initial value \((b)\) of the linear function based on the table provided, we can perform the following steps.

Given the data points:

1. \((-5, 8)\)
2. \((-7, 14)\)

Step 1: Calculate the Rate of Change (m)

The rate of change can be calculated using the formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the values from the points:

\[ y_1 = 8, \quad y_2 = 14, \quad x_1 = -5, \quad x_2 = -7 \]

Now substitute into the formula:

\[ m = \frac{14 - 8}{-7 - (-5)} = \frac{6}{-7 + 5} = \frac{6}{-2} = -3 \]

Step 2: Determine the Initial Value (b)

The initial value \(b\) is the \(y\)-intercept of the linear equation \(y = mx + b\). To find \(b\), we can use one of the data points along with the value of \(m\).

Using the point \((-5, 8)\):

\[ y = mx + b \implies 8 = -3(-5) + b \]

Calculating:

\[ 8 = 15 + b \implies b = 8 - 15 = -7 \]

Conclusion

We have found:

• The rate of change: \(m = -3\)
• The initial value: \(b = -7\)

Therefore, the answer is:

m = -3, b = -7 (first option).