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.