Tangent Lines

Suppose we have the quadratic function \(Q(x) = \frac {1}{2} (x - 2)^2 + 1\).

Then its graph is a parabola. The point \(\left ( 3, \frac {3}{2} \right )\) is on the parabola.

The line \(y=x-\frac {3}{2}\) goes through the point \(\left ( 3, \frac {3}{2} \right )\).

The parabola and line have a special relationship, which we can see by zooming in.

\(\blacktriangleright \mathcal {Desmos}\): Tangent Line

[ Click on the arrow to the right to expand for a Desmos version of the graph. ]

As you zoom in, the graph slowly looks more and more like the line.

The line does the best job of approximating the graph at the point \(\left ( 3, \frac {3}{2} \right )\).

And, this is the only line that will work for the point \(\left ( 3, \frac {3}{2} \right )\). Any other line, besides this one, will have a permanent (nonzero) angle between the graph and the line.

The graph and this particular line have the same “slope” at the point.

This line is called the tangent line for \(Q(x) = \frac {1}{2} (x - 2)^2 + 1\) at the point \(\left ( 3, \frac {3}{2} \right )\).

Since “slope” is a characteristic of a line and most curves are not lines, curves do not have a slope as we know that word.

However, it is easy to see that curves have something that seems a lot like a slope at points.

As the parabola curve above demonstrates, curves might have different slopes at different points.

We want to quantize this idea of a slope for a curve at a point.

If we can quantize it, then we can build a function for it and use that to measure slopes for our curve. This is the first step toward Calculus. It will take us a while to quantize this idea of slope for a curve into a logical tool.

We will first investigate this idea with quadratic functions and parabolas.

\(\blacktriangleright \mathcal {Desmos}\): Secant Line