Thread: Point of intersection of two curves

1. 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

2. 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 $\displaystyle (2, 7)$.