Find the rate of change and the equation for the following: f(4)=1 and f(8)=7

rate of change= 7-1)/(8-4)=6/4

f(x)=ratechange*x + b
f(4)=6/4 *4 + b or
1=6+b or b=-5

equation: f(x)=6/4 x -5

To find the rate of change and the equation for the given points, we first need to find the slope of the line passing through these two points.

The slope of a line can be calculated using the formula:
slope (m) = (Change in y) / (Change in x)

Given points: (4, 1) and (8, 7)

Change in y = 7 - 1 = 6
Change in x = 8 - 4 = 4
Slope (m) = 6 / 4 = 3/2

The slope represents the rate of change of the function. In this case, the rate of change is 3/2.

Now, let's find the equation of the line using the slope-intercept form: y = mx + b.

We already have the slope (m) as 3/2. To find the y-intercept (b), we can use the equation:
y = mx + b

Using the point (4, 1):
1 = (3/2) * 4 + b
1 = 6 + b
b = 1 - 6
b = -5

Now we have the slope (m) as 3/2 and the y-intercept (b) as -5. Therefore, the equation of the line is:
f(x) = (3/2)x - 5