# Constructing tangent lines to a function

Let us assume I have a parabola, or some kind of arbitary function.

Now, my question is: How can I geometrically construct the tangent line to a part of the function?

Above is just an example graph. I want to know how I can construct a tangent line through A without knowing the expressions =)

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"geometrically construct" - as in ruler/straightedge? I don't believe there's a general method; try looking up the methods for constructing tangents to conics, for instance. – J. M. Feb 8 '12 at 22:59
Whoops, quite the error in my previous comment; I of course intended to say "compass/straightedge". Silly me... – J. M. Feb 8 '12 at 23:23

The slope of the tangent line at $A = (a, f(a))$ is the derivative of the function $f$ at $a$. Approximations can be obtained by using the slopes of secant lines from $A$ to points on the curve near $A$. Better approximations can be obtained by using secant lines between points equidistant from $A$ on both sides. That is, take a small circle centred at $A$, cutting the curve at $B$ and $C$, and draw the line through $A$ parallel to the line $BC$.
Find the derivative of the function you have and plug in your interested point say $A(x1,y1)$ to find the slope of the tangent at that point. Now find the equation of that tangent line using slope point form. And you can plot that line in any way you want...