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).

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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.

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