This is a series of six posts about how I approach topics in Pre-Cal. They represent a sample of what I get the most questions about which always seem to start "how do you..."
Students should be comfortable with function notation. Students should see the division of two functions as a ratio. Students should be able to discuss the difference between an asymptote and removable discontinuity. Students should be able to understand why they occur and how they appear on a graph.
There is a lot here. I talk a lot about algebra, a lot about graphs, and try to cycle between them frequently.
Start where we left off in composite functions. We discussed addition, subtraction, and multiplication but have ignored division. I specify division as the ratio of two independent function and coin the term rational as shorthand. Define some functions f(x) and g(x), both factorable quadratics, and such that the ratio will generate some errors. Have the students create a table of x values from -10 to 10. Have them evaluate f(x) and g(x) for the length of the table. Then evaluate the ratio of f(x) / g(x). They should see a g(x) value of 0 causes a problem. Glenn Waddell has a prepackaged alternative version of this in Desmos.
Graph the ratio of f(x) / g(x) with Desmos (either individually or just you) to provide some explanation for the errors. At this point I'll define asymptote and removable discontinuity.
Now we get into why these errors happen, why some are considered removable, and how you could find their location without a graph. I prepare a number of factorable quadratics as ratios and have them simplify them through factoring to see if any share a binomial in the denominator and numerator. It is tempting to say that if you get something like (x - 3)(x + 4) / (x + 4)(x - 7) that the ratio will fail at -4 and 7 because those are the opposites. It is much better to phrase it as "what value of x will make the binomial zero?" I'm careful to say divide out and not cancel out as well when defining x = -4 as a hole.
Graph a Rational without a Calculator: present them with a ratio made of factorable quadratics, have them identify the location of any asymptotes or removable discontinuities. Use these locations as boundaries. Mark the asymptote with a dotted line and notch a spot on the axis where the hole should appear. Use test points on either side of the asymptote and hole to see if the graph is generally positive or negative. Draw curves that follow that behavior.
Identify a ratio from a graph: Provide some graphs of rational functions. Have them work backwards from the location of the asymptotes/holes to write out the possible denominator. Fill in what's possible based on the presence of a hole. Use the idea of a placeholder in the numerator to signify the lack of additional information. Ex: unknown A / (x + 3)(x - 9) for something with two asymptotes but no holes.
Design a Rational: make up two ratios that demonstrate the characteristics we have seen. Demonstrate a function with multiple asymptotes and one of them should have at least one asymptote and a hole. Label the graph with the corresponding ratio and explain why the features appear where they do.
Students are tested on mechanics: identifying the location of an asymptote or removable discontinuity from a ratio of factorable quadratics/cubics. Students are tested on context by being given a ratio and possible graph of that ratio and have to develop an argument as to whether or not that graph could represent the ratio. A few students did a fantastic job with those questions. This is tested as two SBG topics.