Find the parabola with equation y = ax2 + bx whose tangent line at (1, 1) has equation y = 6x − 5.

Answer 1

The derivative is y' = 2ax + b.

Since the point of tangency is at (1,1) the slope of the tangent line is found by substituting 1 for x. That results in 6 = 2a +b. Note: we know the slope of the tangent line is 6 because the equation of the tangent line is given in slope intercept form with m = 6.

Also we can substitute 1 for x and 1 for y in the original function (since (1,1) must lie on the parabola).

This gives 1 = a + b.

Solve this system to find the values of a and b.

2a + b = 6.

a + b = 1.

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