What is the rate of change and initial value of the linear function modeled by a line passing through the points (0,8) and (3,-1)?

The rate of change is equal to rise over run
= (-1 - 8)÷(3 - 0)
= -9 ÷ 3
= -3

When you say initial value I'm assuming you mean the y-intercept

In that case you first need to find the equation of the line by substituting the gradient and point into the gradient intercept formula

y - 8 = -3(x - 0)
y = -3x + 8

The y-intercept is when x = 0 so,

y = -3(0) + 8
Therefore y = 8