I am a gradudate student teaching college algebra at a larger state school and transformations of graphs of function, i.e.: given the graph of a function $y =f(x)$, what do the graphs $y = f(x) \pm C$, $y = f(x \pm C)$, $y = Cf(x)$, $y = f(Cx)$, $y = -f(x)$, and $y = f(-x)$ look like compared to the original graph? This is a topic that I always struggle to come up with concrete “real world” applications of. My audience is typically freshman who are looking to get the class out of the way, so they can move on to business calculus or so they can be done with math altogether. Consequently, teaching material through a real scenario is a good way to hold my students attention.
However with this topic, I have a hard time coming up with good examples. One such example I gave was the following: I presented them with the graph $y = f(x)$, where $f(x)$ is the function that takes in the number of miles/hour over the speed limit a driver is going and returns the value of the corresponding fine for speeding in a particular city. I then asked them how the graph would change as well as how they could modify $f(x)$ using these elementary transformations if (a) the city decides to crack down on speeding and raise all fines amount by \$50 ( $y = f(x) + 50$) and (b) the city decides to be more lenient and give drivers a 5 mph “buffer” ($y = f(x-5)$) – meaning that a driver would not receive a fine unless they exceeded the speed limit by more than 5 mph.
Does anyone else have better examples that are rooted in a real scenario?