The graph of f(x) passes through the point (0,2). The slope of f at any point P is three times the y-coordinate of P. Find f(1).

We are given (0,2) as a point on a line.  We are told that
the slope of the line at any point P is three times the y-coordinate of P.  We are asked
to find f(1).

We must first find the equation of the
line.

We will use the slope intercept equation to find the
y-intercept for the line.

y = mx +
b

Substitute the given point and slope (m) into the
equation to find b.  The slope is 3 times the y value of the given
point.

=> m =
3(2)

=> y = mx +
b

=> 2  = 6(0) + b  (substituting the slope and
given point)

=> 2 =
b

We now substitute the known slope of 6 and the
y-intercept into the slope intercept equation.

=> y
= 6x + 2

We have been asked to find
f(1).

=> y = 6(1) +
2

=> y =
8

The value of  f(1) is
8.