dg.differential geometry - How can one best link a circle to a straight line by an arc of continuously varying curvature?

Are there well known plane curves with equations of the form y=f(x), which specify an arc s in the Cartesian plane E-having distinct end points P1,P2-so that the following conditions may be satisfied?

Let R be a given positive real number. Let C be a circle in E having radius R, which does not intersect
the x-axis and has its center on the positive y-axis. f(x) is a strictly decreasing function of x and
the curvature of its graph s is a continuous and strictly decreasing function of arc length. P1 is the only
intersection point of s and C. s and C have the same tangent at P1 and the curvature of s at P1 is 1/R.
s is tangent to the x-axis at P2 and its curvature at P2 is zero.