Today we are going to extend solving systems of linear equations to non-linear equations. I begin the lesson by giving students a linear system solve. As students work I identify the methods used by students. Most students either rearrange the equations to put the equations into a calculator or solve by elimination (also called linear combination).
After a few minutes I have students share their process in solving. I give students the names of the 3 methods learned in Algebra 1 and 2. Namely elimination, graphing and substitution. I ask students to identify which process is demonstrated on the board.
Since I rarely have students who do substitution on this problem we go through the substitution method on the problem. I ask students why substitution is not the most efficient method for this system.
Now that we have reviewed solving systems I give students a problem with one equation being non linear. I ask students to decide how they can solve the problem.
Students discuss ideas with each other and some start to work on the problem. After a couple of minutes I have students share ideas. Some solve the second equation for y and then substitute. Other try elimination by subtracting the second equation from the first.
Students that have started working the problem put their process on the board for discussion. To help students see how the graphs intersect, I graph (page 2) the functions in Desmos. Desmos allows the equations to be typed without rearranging which helps in the class. We identify the points of intersection and use the graphs to verify what happens in the algebra.
Some students comment on how these equations intersect twice. My comment back is "That is interesting how many times can 2 linear equations intersect? Why do these intersect twice? Will a quadratic and a line always intersect two times?"
As the students discuss the questions I use graphs to demonstrate that the graphs may not intersect or may intersect at only 1 point. I ask "could a linear function and a quadratic function intersect at more than 2 points?" Students discuss how 2 would be the maximum because of the end behavior of the functions. "Are there any mathematical figures that could intersect at more than 2 points?" I give students a hint that the equations do not have to be functions. I have students sketch shapes that may intersect more than twice.
I want students to know there is a reason find points of intersection of equations. A place the many of my students will use today's topic is in calculus. When calculus students need to find the area between curves they have to find the points of intersection. The students also determine which function is larger for the interval.
I explain this idea to the students and use the problem we just solved to answer 2 questions.
When students identify the interval we discuss how this is given with the x-values since we need to know where not the value. The graph makes it easy to find the function that has larger values in the interval. I ask students how we could determine which has larger values if we did not have a calculator. I let students brainstorm. Some will say sketch the graphs. "Okay how will you sketch the graphs?" Students comment on finding point in the interval. "So if I find points in the interval for each function can I use those points to determine which function is larger?" Students begin to realize they can pick an x in the interval evaluate each function to determine which function is larger.
I now give students some problems to work independently. The first 3 problems are straight forward and should be solvable for any students. The fourth problem give student a challenge since there are 3 points of intersection so the upper and lower functions change. This assignment allows students to practice solving non linear systems and prepare for a skill needed for calculus.
As class ends I ask students to determine the number of intersections for 2 parabolas.
The equations are in a family of functions so they will not intersect. At first students think they will intersect twice. I ask the students to verify the answer. When they try to solve the system using algebra they have a no solution result. I ask "why do you think these 2 equations will not intersect? Can you find another equation that will not intersect these 2 equations?"
Some students notice that the 2 equations are just different by a vertical shift. Students comment on the slope being the same. I ask what do you mean slope these are not lines. What term is better in this situation. I help students come up with the term "rate of change" is the same on both functions so as x increases from 1 to 2 or 2 to 3 the y increases by the same amount on each equation. The students realize they can find an infinite number of equations that do not intersect by changing the constant on the function.