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=
.
To find the rate of change (m) and the initial value (b) of the linear function, we can use the formula for the slope of a line:
m = (y2 - y1) / (x2 - x1)
Let's plug in the values from the table:
For the first pair of points (x1, y1) = (1, 4) and (x2, y2) = (-3, 8):
m = (8 - 4) / (-3 - 1)
m = 4 / -4
m = -1
So, the rate of change is -1.
To find the initial value (b), we can use the point-slope form of a linear equation, which is given by:
(y - y1) = m(x - x1)
Since we have the value for m and one set of coordinates (1, 4), we can rearrange the equation to solve for b:
(y - 4) = -1(x - 1)
y - 4 = -x + 1
y = -x + 5
Comparing this equation to the slope-intercept form (y = mx + b), we can see that b = 5.
So, the initial value is 5.