A function passes through the points (2,9) and (7,34). What are its rate of change and initial value?
Rate of change

The rate of change: 1/5

Initial value: (0,-1)

Step-by-step explanation:

The rate of change is easily 5/25, we just have to simplify it to 1/5.

To get the intial value, just subtract (2,9) by the rate of change twice.

1 number

if we use a graphing calculator, there is only one such number that satisfies the conclusion of the mean value theorem, that is, a number such that

where the right-hand side is the average rate of change of f on the interval [0.5,0.75], which is the slope of the secant line.

in the attached graph, the black line is the secant line which has its slope as the average rate of change of f on [0.5,0.75], while the orange line is the only tangent line to the graph of f in the interval that satisfies the mvt conclusion.