Find an equation of the line that is tangent to the graph of f and parallel to the given line.

Function Line
f(x) = 2x^2 6x – y + 4 = 0

I know i have to find derivative 1st which will be f'(x)=4x but then how to i find equation do i plug 6 in for x so 4(6) would it be 6x-24???

did you look at the answer by

MATH - Reiny, Monday, February 13, 2012 at 6:07pm ?

Yes i did but that was wrong so i am confused now...

Never mind I got it thnks

To find the equation of the line that is tangent to the graph of f and parallel to the given line, you need to follow these steps:

1. Find the derivative of the function f(x) to determine the slope of the tangent line. In this case, you correctly found that f'(x) = 4x.

2. Since the tangent line is parallel to the given line, it will have the same slope. Look at the equation of the given line, which is 2x^2 - 6x - y + 4 = 0. Since the coefficient of x is 2 in the equation, the slope of the given line is also 2.

3. Set the slope of the tangent line (found in step 1) equal to the slope of the given line (found in step 2) to find the x-coordinate of the point of tangency. So, 4x = 2.

4. Solve the equation from step 3 to find the x-coordinate of the point of tangency. Divide both sides of the equation by 4: x = 2/4 = 1/2.

5. Plug the x-coordinate of the point of tangency (from step 4) into the original equation of the function f(x). f(1/2) = 2(1/2)^2 - 6(1/2) - y + 4 = 1/2 - 3 - y + 4 = -7/2 - y + 4 = -3/2 - y = 0. Simplifying further, we get y = -3/2.

Therefore, the point of tangency is (1/2, -3/2).

6. Now that we have the point of tangency, we can use the point-slope form of a linear equation to find the equation of the line. The equation is given by y - y1 = m(x - x1), where (x1, y1) is the point of tangency and m is the slope. We know that the slope of the tangent line is 2 (since it is parallel to the given line).

Substituting the values into the equation, we get:

y - (-3/2) = 2(x - 1/2)
y + 3/2 = 2x - 1
y = 2x - 1 - 3/2
y = 2x - 2.5

Therefore, the equation of the line that is tangent to the graph of f and parallel to the given line is y = 2x - 2.5.