The intersection point between graphs

Consider two functions, e.g.

f(x)=x^2

and

g(x)=2x^2-x-1

If we draw the graphs, we see that they intersect at two points $A$ and $B$ (see below). How can we find the coordinates of these points?

Let's start by noting once more that the values along the $x$ -axis represent input values to the machines $f$ and $g$ , and the values along the $y$ -axis are their output values. For example, if we choose input $x=0.5$ , machine $f$ has the output

y=f(0.5)=0.5^2=0.25

and machine $g$ has the output

y=g(0.5)=2\cdot 0.5^2-0.5-1=-1

giving raise to the two points $P_1(0.5\vert 0.25)$ and $P_2(0.5\vert -1)$ on the graph of $f$ and $g$ . In particular note that although they have the same input, the output (or $y$ -coordinate) of the machines are different. This, of course, is to be expected, as they are different machines. Note, however, that at the point of intersection, $A$ , machine $f$ and machine $g$ produce for the same input the same output or $y$ -coordinate, and this is also true for point $B$ .

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So to find the $x$ -coordinate of a point of intersection between $f$ and $g$ , we have to search for those input values for $f$ and $g$ which produce the same output:

\boxed{x \text{ is } x \text{-coordinate of intersection point if } f(x)=g(x)}

As $f(x)=x^2$ and $g(x)=2x^2-x-1$ , finding the $x$ -coordinate amounts to solving the equation

x^2 = 2x^2-x-1

At the moment we cannot solve this particular type of equation, but it turns out that this equation has the two solutions $x_1=-0.618$ and $x_2=1.618$ (verify it!). So the intersection point $A$ has the $x$ -coordinate $-0.618$ , and $B$ has the $x$ -coordinate $1.618$ . How do we find out $y$ -coordinates of these points? Well, as $A$ is on the graph of $f$ , the $y$ -coordinate of $A$ must be

y=f(-0.618)=(-0.618)^2 = 0.382

and the $y$ -coordinate of $B$ must be

y=f(1.618)= (1.618)^2= 2.618

As $A$ and $B$ are also on graph $g$ , we could find their $y$ -coordinate by using $g$ as well:

y=g(-0.618)=2\cdot (-0.618)^2-3\cdot (-0.618)=0.382

and the $y$ -coordinate of $B$ must be

y=g(1.618)=2\cdot (1.618)^2-3\cdot 1.618= 2.618

Thus, the points of intersections have the coordinates $A(-0.618 \vert 0.382)$ and $B(1.618 \vert 2.618)$ .

Exercise 1

Find the intersection point(s) between the graphs of $f$ and $g$ :

$f(x)=2x$ , $g(x)=-0.5x+1$
$f(x)=1$ , $g(x)=x^2$
$f(x)=1$ , $g(x)=\frac{25}{x^2}$
$f(x)=x^2+1$ , $g(x)=-2x^2$

Solution

Find $x$ with $f(x)=g(x)$ , then solve the equation to get the $x$ -coordinates of the intersection points.

$S(0.4\vert 0.8)$
$S_1(-1\vert 1), S_2(1\vert 1)$
$S_1(-5 \vert 1), S_2(5 \vert 1)$
no solution, so no intersection point!