Conics, then and now

Back in 1995, high school students learned about conics. A typical test question about them looked something like this:

Give the equation, in standard form, of the conic whose graph is pictured below:

While this question still resides squarely outside the domain of rocket science, it tests a number of skills that are required to solve it correctly and efficiently. Students must know what “standard form” means in the context of equations of conics. They must know the equations of the rectangular hyperbolas x^2-y^2=1 and y^2-x^2=1, and they must be able to identify which of those two has a vertical transverse axis and which has a horizontal one. They must know how to use the asymptotes, whose graphs are given in the picture, to figure out the stretches or compressions applied to the rectangular hyperbola, and they must know how those transformations are reflected in the hyperbola’s equation. They must know how the pictured hyperbola’s centre at (2,-4) factors into the hyperbola’s equation. Only by using all of that information (or by developing the equation of a hyperbola from first principles) can students can find that they have a hyperbola with equation ((y+4)^2)/9-((x-2))^2/4=1 on their hands.

All in all, a decent question.

In 2005, students still learn about conics, but times have changed. The following question appears in a test on conics given across BC high schools:

Which of the following is the equation in standard form of the conic whose graph is shown below?

a) ((y+4)^2)/9-((x-2)^2)/4=1
b) ((y-4)^2)/9-((x+2)^2)/4=1
c) ((y+4)^2)/4-((x-2)^2)/9=1
d) ((y+4)^2)/4-((x-2)^2)/9=1

This question tests a different set of skills: it tests whether students can plug the equations of four different hyperbolae into their fucking graphing calculators and pick out the one that looks like the one in the picture.

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