Doug C. answered • 10/22/19
Math Tutor with Reputation to make difficult concepts understandable
The point that the two functions share is (1,1)--find that by letting x = 1 in the 2nd function.
Find y' of the 2nd function to determine the slope of the tangent line at (1,1) along with the equation of the tangent line--using quotient rule.
You will find that the derivative has a value of zero when x = 1, which means the slope is zero at (1,1) or horizontal tangent line y = 1.
Turning to the parabola, it must be the case that the point (1,1) is the vertex of the parabola (vertex has the horizontal tangent).
Also when x = 1 and y =1 we get 1 = a + b. We need a 2nd equation involving "a" and "b". We could use the fact that the axis of symmetry is given by the equation x = -b/2a.. That means -b/2a = 1. Another possibility is to find the derivative of the parabola: y' = 2ax + b. When x =1 we want y' to equal 0. So 2a+b = 0. Either gives b = -2a. Replace "b" in the first equation with -2a to get the value of "a". Then find b.
Attach the following to the URL for the Desmos web site to see the answer along with both graphs:: /calculator/mfuzix862x
