![]() $f(Cx)$: Convert the units of the input of the function by a scalar, e.g.We are thankful to be welcome on these lands in friendship.$f(-x)$: Instead of measuring time "after" an event measure time "before" an event.If $A(t)$ is the altitude of a climber at time t, write a function describing their height above the tree line at time t." "The tree line is _, and located 10,000 feet in altitude. $f(x)\pm C$: Surcharges and flat fees.Tax deductions (taxable and non-taxable income). ![]() $f(x\pm C)$: Measuring traffic density in "hours after 5am" instead of "hours after midnight". ![]() Some other specific ways to incorporate each individual transformation could include: Of course, these are intentionally challenging examples that are intended to pack as much possible into one problem. What will the graph of your new function look like? Then, adding $y_o$ shifts the graph $y=-gt^2/2$ up by $y_o$ and in this discussion $y_o$ has the significance of being the initial condition.Ī more ambitious formula to explain would be $y = y_o v_ot-\fracC(30m) 32)$. The significance is that $g$ is the acceleration due to gravity. If we start with $y=t^2$ then multiplying b $-g/2$ stretches the graph and flips it over. To, give a less exciting, but very much applied example consider: There is much to explore here.Įdit: as Dave Renfro pointed out, I am using forbidden trigonometry. Or, if we add time I think you can get beats. Of course, adding graphs has interesting interpretations in terms of constructive and destructive interference. I happen to have this graph of a solution to the wave equation sitting around. Other more complicated wave graphs could be studied. To see it, I think setting $0 \leq x-vt \leq 2\pi$ might be helpful, then $vt \leq x \leq 2\pi vt$ and if we animate $t$ we can actually see the wave travel. This is a right-moving wave if we assume $v>0$. In particular we shift the $y = A \sin(x)$ graph $vt$-units to right to form $y = A \sin(x-vt)$. If we fix $t$ then the term $-vt$ is just some phase-shift and we can see the graph is just a horizontal shift of the sine wave. Here $A$ gives the amplitude of the wave and it tells us from a graphical perspective how much the unit-sine wave is vertically stretched. My favorite example of a graphical transformation is waves. 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.ĭoes anyone else have better examples that are rooted in a real scenario? 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. However with this topic, I have a hard time coming up with good examples. Consequently, teaching material through a real scenario is a good way to hold my students attention. 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. This is a topic that I always struggle to come up with concrete “real world” applications of. Given the graph of a function $y =f(x)$, what do the following graphs look like, compared to the graph of $y = f(x)? I am a graduate student teaching college algebra at a larger state school, and currently I'm covering transformations of graphs of function, i.e.:
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