Points of intersection

In a nutshell

Points of intersection are coordinates where two curves cross or touch each other. The $$x$$ coordinate of points of intersections can be found by solving the curves' equations as simultaneous equations. This can be done by equating the relevant equations and solving for $$x$$.​

Finding points of intersection

Points of intersection between two curves, $$y=f(x)$$ and $$y=g(x)$$, can be estimated by sketching the curves on the same diagram. The exact coordinates can be found from the solution to $$f(x)=g(x)$$. ​

​​Example 1

Sketch the functions $$y=(x)(1-2x)$$ and $$y = (x)(x-3)(x+1)$$. How many points of intersection are there?

Identify the degree and sign of the polynomial, $$y\text-intercept$$​ and the roots. 

Use this information to sketch the curves.

Use the sketch to identify the number of points of intersection between the curves.

There are $$\underline 3$$ points of intersection.

Example 2

What are the coordinates of the points of intersection between the curves $$f(x)=(x)^2(x-4)$$ and $$g(x) = (x)(2x-9)$$?

Write $$f(x)=g(x)$$ in terms of $$x$$.

$$(x)^2(x-4) = (x)(2x-9)$$​​

Expand the brackets. 

$$x^3-4x^2 = 2x^2 - 9x$$​​

Collect the $$x$$ terms on one side. 

$$x^3 - 6x^2 +9x = 0$$​​

Completely factorise the equation. 

​$$\begin{aligned} &x(x^2 - 6x +9) &= 0 \\& x(x-3)^2 &=0 \end{aligned}$$​​

Solve for $$x$$.

$$\begin{aligned} x&=0 \\ \\ x-3 &= 0 \\ x &=3 \end{aligned}$$

Substitute $$x$$ into $$f(x)$$ or $$g(x)$$ to find the $$y$$ coordinate.

​$$\begin{aligned}g(0) &= (0)(2(0)-9) = 0 \\ g(3) &=(3)(2(3)-9)=(3)(-3) = -9 \end{aligned}$$​​

Therefore, $$f(x)$$ and $$g(x)$$ intersect at the coordinates: $$\underline{(0,0)}$$ and $$\underline{(3,-9)}$$.

Note: If an $$x$$ coordinate is a point of intersection, the $$y$$​ coordinate will be the same if you substitute the $$x$$ value into the equation of either curve.