# Point of intersection of two curves

• Aug 7th 2011, 03:30 AM
celtic1234
Point of intersection of two curves
Hi
I am struggling with this method-my book does not explain it very well.
Can anybody recommend a method to follow? or maybe an example on line?
thanks
John
• Aug 7th 2011, 04:32 AM
Bacterius
Re: Point of intersection of two curves
Let the two curves be defined by the equations \$\displaystyle y_1 = f(x_1)\$ and \$\displaystyle y_2 = g(x_2)\$. The points of intersection of these two curves is defined as the solutions to the conditions \$\displaystyle x_1 = x_2\$ and \$\displaystyle y_1 = y_2\$. Substituting the curve equations into the previous equation, we get \$\displaystyle f(x_1) = g(x_2)\$, and since \$\displaystyle x_1 = x_2\$, this is equivalent to \$\displaystyle f(x_1) = g(x_1)\$. Let \$\displaystyle x_1 = x\$ for simplicity, and it follows that solving \$\displaystyle f(x) = g(x)\$ is equivalent to finding the x-coordinates of all the intersections between the two curves.

So for instance if you want to find the points of intersection of the equations:

\$\displaystyle y = x^2 + 2x - 1\$
\$\displaystyle y = 2x + 3\$

You will want to solve \$\displaystyle x^2 + 2x - 1 = 2x + 3\$, which simplifies to \$\displaystyle x^2 - 4 = 0\$. The solutions are therefore \$\displaystyle x = \{-2, 2\}\$, and substituting back these x-values into either of the curve equations will give you the y-coordinates of the points of intersection, which are \$\displaystyle y = \{-1, 7\}\$ respectively. Thus the points of intersection are \$\displaystyle (-2, -1)\$ and \$\displaystyle (2, 7)\$.
• Aug 8th 2011, 12:23 PM
celtic1234
Re: Point of intersection of two curves