Using technology to solve problems with graphs I
Often when we are modelling real life scenarios, we do so by using different functions to represent the situation we're investigating.
An example you've no doubt seen before is when you compare the fees charged by two electricians and you end up comparing two linear functions and their graphs. The aim might be to calculate when both electricians charge the same fee.
Another example you might he familiar with is when we want to examine the value of two investments, and we model each using an exponential function. The aim might be to calculate when both investments are worth the same.
We can always calculate these answers using algebraic techniques, but here we want to focus on the use of technology.
Are you solving for $x$x, or finding the point of intersection?
Before we launch into the various technologies, we need to get something straight - are we only interested in the $x$x value of where the two graphs intersect, or are we interested in finding the coordinate of intersection?
How can you tell? Easy! It's all in the wording.
If you asked "Find where $f\left(x\right)$f(x) and $g\left(x\right)$g(x) intersect" then this means you want coordinate(s) of intersection, not just the $x$x value.
If you asked "Solve $f\left(x\right)=g\left(x\right)$f(x)=g(x)" then this means you only want the $x$x value of intersection. For example, if $f\left(x\right)=x^2+9$f(x)=x2+9 and $g\left(x\right)=x+5$g(x)=x+5 then writing $f\left(x\right)=g\left(x\right)$f(x)=g(x) can be written as $x^2+9=x+5$x2+9=x+5.
See how the only variable in the equation is $x$x? That means we're only interested in the value(s) of $x$x.
Graphing Technologies
We'll examine the following different technologies in this chapter. You should become familiar with at least one of these, and in particular the form of technology recommended by your school.
- Geogebra (Free, web-based technology)
- TI-Nspire (A popular graphing CAS calculator)
Geogebra
Below is a video showing you the basics of graphing with Geogebra, a very powerful and excellent free online tool.
TI-Inspire
Check out these videos made by our very own Daniel O'Kane.
Finding the intersection of two graphs
Another way to find the intersection of two graphs is to solve the two equations simultaneously. This gives you both the $x$x value and the $y$y value of each point of intersection. You must decide whether you wanted just the $x$x values or the coordinate of intersection.
Solving two simultaneous equations
Worked Examples
QUESTION 1
Using the graphs of $y=2x-7$y=2x−7 and $y=-2x+1$y=−2x+1, find the solution(s) of the equation $2x-7=-2x+1$2x−7=−2x+1.
QUESTION 2
By graphing $y=3x^2-2x-40$y=3x2−2x−40 and $y=-x^2+2x+8$y=−x2+2x+8 on a graphing calculator or otherwise, find the solutions of the equation $3x^2-2x-40=-x^2+2x+8$3x2−2x−40=−x2+2x+8. Write both solutions on the same line separated by a comma.
QUESTION 3
By graphing $y=2x+4$y=2x+4 and $y=4^x+1$y=4x+1 on a graphing calculator or otherwise, find the solution(s) of the equation $2x+4=4^x+1$2x+4=4x+1.
State all solutions on the same line, separated by a comma. Give your answer correct to two decimal places.