Intersections of graphs by other methods II

We recall that the points of intersection of two curves $y=f\left(x\right)$y=f(x) and $y=g\left(x\right)$y=g(x) are found by setting $f\left(x\right)=g\left(x\right)$f(x)=g(x). 

For example the single intersection point of the two lines $y=mx+b$y=mx+b and $y=px+q$y=px+q can be found by setting $mx+b=px+q$mx+b=px+q. Gathering like terms together onto one side shows $\left(m-p\right)x+\left(b-q\right)=0$(m−p)x+(b−q)=0 and thus the $x$x- value of point we are searching for is given by $x=-\frac{b-q}{m-p}=\frac{q-b}{m-p}$x=−b−qm−p​=q−bm−p​. 

Note that this last result is not a formula to be learnt, but rather a concept to be understood. We learn two things from this algebra. Firstly, the lines do not intersect if $m=p$m=p, and secondly there is only one possible point of intersection between any two two lines.

But we can put it into perspective through and example.

Example 1

Find the single intersection point of the two lines $y=2x+3$y=2x+3 and $y=x-2$y=x−2.

This can be found by setting $2x+3=x-2$2x+3=x−2. Gathering like terms together onto one side shows $\left(2-1\right)x+\left(3--2\right)=0$(2−1)x+(3−−2)=0 and thus the $x$x-value of point we are searching for is given by $x=-5$x=−5. 

 

We can extend the same idea to higher degree polynomials. A quadratic function $y=px^2+qx+r$y=px2+qx+r and a line $y=mx+b$y=mx+b can only intersect in at most two places, because there can only ever be two solutions to the equation $ax^2+bx+c=mx+b$ax2+bx+c=mx+b. The same reasoning informs us that there can never be any more than $3$3 intersections between the cubic function and the quadratic function. 

With other function forms however, this is not so clear. For example, the exponential curve given by $y=\frac{1}{2}\left(2^x-x\right)$y=12​(2x−x) and the absolute value function given by  $y=\left|x+1\right|$y=|x+1| intersect in three points as shown in this graph:  

The intersection points were found using  computer technology, and are given as approximately  $\left(-2.215,1.215\right)$(−2.215,1.215), $\left(-0.417,0.583\right)$(−0.417,0.583) and $\left(3.717,4.717\right)$(3.717,4.717).

There are various clever mathematical methods that the technology uses to find these intersections. One such method is called the Newton-Raphson method which involves concepts in the calculus.

We can however use a plotting strategy to locate the approximate positions of intersections.

Without actually graphing the functions, we could determine, say, $11$11 function values, for both functions, for integer values of $x$x from say $x=-5$x=−5 to $x=5$x=5. 

This table shows the $5$5 function values for $x=1,2,3,4,5$x=1,2,3,4,5 on both functions:

x value	1	2	3	4	5
$y=\frac{1}{2}\left(2^x-x\right)$y=12​(2x−x)	0.5	1	2.5	6	13.5
$y=\left|x+1\right|$y=|x+1|	2	3	4	5	6

Looking at the table only, the relative sizes of the $y$y - values of the two functions change around between $x=3$x=3 and $x=4$x=4. This tells us that there is an intersection point within the interval $3.$3.

To progress further we could divide this shorter interval up and check again for a change in relative sizes. Continued application of this process can lead to better and better approximations of the points of intersections.

Worked Examples

QUESTION 1

QUESTION 2

QUESTION 3