Question

Find the rate of change and the initial value of the linear function given as a table of values.
x y
1 4
-3 8
The rate of change is m=, and the initial value is b=.

Answer 1

To find the rate of change (m) and the initial value (b) of a linear function, we can use the formula for finding the slope (rate of change) of a line given two points (x₁, y₁) and (x₂, y₂):

m = (y₂ - y₁) / (x₂ - x₁)

Let's use the values from the table to find the rate of change (m):

x₁ = 1, y₁ = 4
x₂ = -3, y₂ = 8

m = (8 - 4) / (-3 - 1)
m = 4 / -4
m = -1

So, the rate of change (m) is -1.

To find the initial value (b), we can substitute one of the points into the slope-intercept form of a linear equation:

y = mx + b

Let's use the first point (1, 4):

4 = -1(1) + b

Simplifying the equation:

4 = -1 + b
b = 4 + 1
b = 5

So, the initial value (b) is 5.

Therefore, the rate of change (m) is -1 and the initial value (b) is 5.