Question

Use the table to answer the question

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.

The rate of change is m = ______ , and the initial value is b = ________

Answer 1

you sure

Answer 2

To find the rate of change (m) and the initial value (b) of the linear function from the given table of values, we can follow these steps:

1. **Calculate the Rate of Change (m)**:
The rate of change (slope) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using the points from the table:
- Point 1: \((1, 4)\) → \(x_1 = 1, y_1 = 4\)
- Point 2: \((-3, 8)\) → \(x_2 = -3, y_2 = 8\)

Substituting the values into the formula:
\[
m = \frac{8 - 4}{-3 - 1} = \frac{4}{-4} = -1
\]

2. **Find the Initial Value (b)**:
The initial value (y-intercept) is the value of \(y\) when \(x = 0\). Since we do not have a point where \(x = 0\), we can use the slope-intercept form of a linear equation:
\[
y = mx + b
\]
We can choose one of the points and substitute the values to solve for \(b\). Let's use the point \((1, 4)\):
\[
4 = -1(1) + b
\]
Simplifying:
\[
4 = -1 + b \\
b = 4 + 1 = 5
\]

Now we have the values:
- The rate of change \(m = -1\)
- The initial value \(b = 5\)

Thus, the answers are:
The rate of change is \(m = -1\), and the initial value is \(b = 5\).

Answer 3

Let’s recheck the calculations just to be sure.

To find the rate of change (m) and the initial value (b):

1. **Given points**:
- \((x_1, y_1) = (1, 4)\)
- \((x_2, y_2) = (-3, 8)\)

2. **Calculate the Rate of Change (m)**:
The formula for the rate of change (slope \(m\)) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Plugging in the values:
\[
m = \frac{8 - 4}{-3 - 1} = \frac{4}{-4} = -1
\]

This part is correct. So, **the rate of change (m) is -1**.

3. **Finding the Initial Value (b)**:
We now need to find the linear equation in the form:
\[
y = mx + b
\]
Using the slope \(m = -1\) and one of the points. Let’s use the point \((1, 4)\):
\[
4 = -1(1) + b
\]
\[
4 = -1 + b
\]
Adding 1 to both sides:
\[
b = 4 + 1 = 5
\]

This part is also correct. So, **the initial value (b) is 5**.

### Conclusion:
The calculations confirm that:
- The rate of change (m) is **-1**.
- The initial value (b) is **5**.

If you have further doubts or there’s specific information you want to double-check, please let me know!