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