Ideas in Geometry/Instructive examples/Lesson 4: Algebraic Geometry

The equation of a line is L(x)=mx+b, where m is the slope and b is the y-intercept.

The equation of a parabola is P(x)=ax^2+bx+c.

A line that is tangent to a parabola goes through the parabola at exactly one point.

To find P(x)=L(x), we can set P(x)-L(x)=0. This equation will have 2 roots because the answer will be a quadratic since a quadratic minus a line equals a quadratic of some sort. THESE ROOTS MUST BE THE SAME, so P(x)-L(x)=(x-c)^2.

Example 1: Find the line tangent to y=x^2-3x+1 at the point x=3.

Solution 1:

P(x)-L(x)=(x-c)^2

x^2-3x+1-L(x)=(x-3)^2

-3x+1-L(x)=-6x+9

L(x)=8-3x

Example 2: Find the line tangent to y=x^3-3x^2+4x-1 at the point x=0.

Solution 2: x^3-3x^2+4x-1-L(x)=x^2(x-c)

4x-1-L(x)=0

L(x)=4x-1