Tangent of Parabola Algebraically

I. Secant Line and Two Generic Points

Let's examine the most basic parabola possible : y = x² and see what happens, algebraically, when the secant becomes the tangent. This approach is known as the delta method.


Why are we doing this?
Because we can use some basic algebra (the slope through two points on the secant) to examine what happens two our two points, which we will call A and B, as they approach each other to become the tangent of the parabola .

In other words, we can figure out the tangent line, algebraically, by looking at the secant for two generic points : point A which has coordinates (x, x²) and B with coordinates (x +Δx, (x +Δx)²) . We are going to examine what happens when these two points get closer and closer together and approach becoming the tangent. Maybe you now see why this is called the delta method--it's all about what happens to Δx ( "delta x").

Picture of Secant Line and Points A and B

Point A has coordinates (x, x2) (Remember if parabola is y = x2, then for any x the corresponding y value is x2) Point B represents a new point that has been moved by Δx and therefore it's new x-coordinate is x +Δx it's new y-coordinate is its x-coordinate squared which is (x +Δx)2

II The Slope of the Secant Line

We can find the slope of Points A and B using our slope of a line formula.

The prior work is just an algebraic of substituting Point A's and Point B's coordinates into the slope formula.

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So how does this relate to the tangent?

The slope of the tangent line is the limit as Δx → 0 (i.e. the limit as 'delta x' approaches 0) of the slope of the secant line : 2x + Δ x . In other words, the tangent line occurs when Points A and B move to the same spot, that is, after all, what limit as Δx → 0 means-- it means the distance("delta") between the two points approaches zero.


As the sequence of images above shows, the tangent line occurs as Δx → 0 and we can now revisit the slope of our secant line, which, if you remember, is 2x + Δx, yielding a tangent line whose slope is 2x +0 or just 2x.

Therefore the equation of a tangent line through any point on the parabola y =x² has a slope of 2x

Generalized Algebra for finding the tangent of a parabola using the Delta Method
If A (x,y) is A point on y = f(x) and point B (x + Δx , y +Δy) is another point on f(x) then

• y = f(x)
• y + Δy = f(x +Δx)
• Δy = f(x +Δx) - y
• 
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