Algebra II: Hidden Intersections of Curves
When solving systems of equations, you have several options at your disposal for finding those common solutions. Linear systems can be solved by hand, algebraically, using elimination, or substitution — often with several applications of the procedure, depending on how many variables are involved. Other options with linear equations are matrices; the matrices also can be done by hand or with a graphing calculator or computer spreadsheet.
Nonlinear systems of equations present different types of challenges. You may have the equations of two functions whose graphs intersect in one, two, three, or more places — or nowhere at all! It's always good to have a general idea as to what you're going to find as you work through the solution — planning is everything.
Consider the graphs of the functions y = x2(x – 3)2 and y = –x2(x – 3)2. They're both fourth-degree polynomials with intercepts at (0, 0) and (3, 0). If you look at their graphs on a graphing calculator, you see a W-shaped curve and an M-shaped curve — mirror images over the x-axis. They share intercepts, so the solutions of the system of equations involving those two functions should be those intercepts. This is the "planning ahead" — considering what you know about the behavior of the curves. If you solve these systems algebraically, you also get the two solutions that x = 0 and x = 3. So, what's the problem? The problem arises if you depend solely on your graphing calculator to find any solutions.
First, you have the "Solver" option that can be used to find the solutions or intersections of curves. You put the two function equations in the graphing menu under y1 and y2. Then, under the Solver prompts, you have the calculator find solutions for 0 = y1 – y2. The calculator fails! It can't find either of the solutions. The response is "No sign change." Because of the algorithm used by graphing calculators, they can't find solutions when the graphs of the functions just touch and don't cross one another.
Your other option using the graphing calculator is to actually look at the graph. First, it will be obvious that the two curves have points in common. But when you use the Calculate/Intersect command, again, your calculator will tell you that there's no solution — no sign change.
You're smarter than the calculator — really you are! You just have to use those smarts and make the calculator work for you. Don't rely on it completely, and plan ahead.