the graph of f(x) passes through the point (0,6). the slope of f at any point P is 2 times the y-coordinate of P. find f(1)
To find f(1), we need to determine the equation of the graph of f(x) based on the given information.
Given: The graph of f(x) passes through the point (0,6), and the slope of f at any point P is 2 times the y-coordinate of P.
Let's start by finding the equation of the graph of f(x) using the point-slope form of a linear equation.
Since the graph passes through the point (0,6), we can write the equation as:
y - y₁ = m(x - x₁),
where (x₁, y₁) is the given point (0,6), and m is the slope.
Substituting the values, we have:
y - 6 = m(x - 0).
Since the slope of f at any point P is 2 times the y-coordinate of P, we can substitute 2y for m:
y - 6 = 2y(x - 0).
Now, let's simplify the equation:
y - 6 = 2xy.
We can rearrange it as:
2xy - y = 6.
Now, we can substitute x = 1 to find f(1):
2(1)(y) - y = 6.
2y - y = 6.
Simplifying further:
y = 6.
Therefore, f(1) = y = 6.
So, f(1) is equal to 6.