Use the table to answer the question.
x
y
2 22
8 25
Determine the rate of change and the initial value of the linear function given here as a table of values.
(1 point)
The rate of change is m=
, and the initial value is b=
x
y
2 22
8 25
Determine the rate of change and the initial value of the linear function given here as a table of values.
(1 point)
The rate of change is m=
, and the initial value is b=
Answers
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Answered by
GPT-4o mini
To determine the rate of change (m) and the initial value (b) of the linear function given the table of values, we use the following information from the table:
- Point 1: (2, 22)
- Point 2: (8, 25)
**Step 1: Calculate the rate of change (m)**
The rate of change (slope) \( m \) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the values from the points:
\[
m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2}
\]
So, the rate of change \( m = \frac{1}{2} \).
**Step 2: Determine the initial value (b)**
The initial value (b) is the value of \( y \) when \( x = 0 \). We can use the slope-intercept form of the linear equation:
\[
y = mx + b
\]
We can use one of the points to solve for \( b \). Let's use the point (2, 22):
\[
22 = \frac{1}{2}(2) + b
\]
Calculating this gives:
\[
22 = 1 + b
\]
\[
b = 22 - 1 = 21
\]
So, the initial value \( b = 21 \).
**Final Answer:**
The rate of change is \( m = \frac{1}{2} \), and the initial value is \( b = 21 \).
- Point 1: (2, 22)
- Point 2: (8, 25)
**Step 1: Calculate the rate of change (m)**
The rate of change (slope) \( m \) is calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substituting the values from the points:
\[
m = \frac{25 - 22}{8 - 2} = \frac{3}{6} = \frac{1}{2}
\]
So, the rate of change \( m = \frac{1}{2} \).
**Step 2: Determine the initial value (b)**
The initial value (b) is the value of \( y \) when \( x = 0 \). We can use the slope-intercept form of the linear equation:
\[
y = mx + b
\]
We can use one of the points to solve for \( b \). Let's use the point (2, 22):
\[
22 = \frac{1}{2}(2) + b
\]
Calculating this gives:
\[
22 = 1 + b
\]
\[
b = 22 - 1 = 21
\]
So, the initial value \( b = 21 \).
**Final Answer:**
The rate of change is \( m = \frac{1}{2} \), and the initial value is \( b = 21 \).
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