Question

Question Use the table to answer the question. Determining the starting value and rate of change from a given value table. x y 1 4 3 8 Find the rate of change and the initial value of the linear function given as a table of values. (1 point) The rate of change is m= , and the initial value is b= .

Answer 1

To determine the rate of change and the initial value from the given table of values, we can use the following approach:

1. **Identify the values:**
- From the table:
- When \( x = 1 \), \( y = 4 \)
- When \( x = 3 \), \( y = 8 \)

2. **Calculate the rate of change (slope):**
The rate of change \( m \) (slope) can be calculated using the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Here, \((x_1, y_1) = (1, 4)\) and \((x_2, y_2) = (3, 8)\):
\[
m = \frac{8 - 4}{3 - 1} = \frac{4}{2} = 2
\]

3. **Determine the initial value (y-intercept):**
The initial value \( b \) corresponds to the value of \( y \) when \( x = 0 \). However, we can find the linear equation using the slope and one of the points. The linear equation can be structured as:
\[
y = mx + b
\]
We can use one of the points to solve for \( b \). Let's use the point \( (1, 4) \):
\[
4 = 2(1) + b
\]
\[
4 = 2 + b \\
b = 4 - 2 = 2
\]

Thus, the rate of change and the initial value are:
- The rate of change \( m = 2 \)
- The initial value \( b = 2 \)

So, you can fill in the blanks:
- The rate of change is \( m = 2 \), and the initial value is \( b = 2 \).