Answer
The equation of the tangent line is: $~~y = x-\frac{1}{4}$
Work Step by Step
$y = x^2$
$y = x^2-2x+2 = (x-1)^2+1$
A line that is tangent to both parabolas has a slope that is equal to the slope of each graph at the point where it meets the graph.
We can see that the second parabola is simply the first parabola translated one unit to the right and one unit up. Therefore, a line that is tangent to both parabolas must have a slope of 1.
We can find the point $x$ where the parabola $y = x^2$ has a slope of 1:
$y = x^2$
$\frac{dy}{dx} = 2x = 1$
$x = \frac{1}{2}$
$y = (\frac{1}{2})^2 = \frac{1}{4}$
The slope of the tangent line is $1$ and the point $(\frac{1}{2},\frac{1}{4})$ is a point on the tangent line. We can find the equation of the line:
$y - \frac{1}{4} = 1(x - \frac{1}{2})$
$y = x-\frac{1}{4}$
