1992 AMC 12 Problem 14

Question

Which of the following equations have the same graph?

\begin{aligned} &\text{I. }y=x-2\\ &\text{II. }y=\dfrac{x^2-4}{x+2}\\ &\text{III. }(x+2)y=x^2-4 \end{aligned}

\mathrm{I}

and

\mathrm{II}

only

\mathrm{I}

and

\mathrm{III}

only

\mathrm{II}

and

\mathrm{III}

only

\mathrm{I},

\mathrm{II}

and

\mathrm{III}

None. All the equations have different graphs

Pay special attention to what happens when

x=-2

Equation II omits one point, while equation III includes an entire extra vertical line

Equation I is the whole line

y=x-2.

Equation II simplifies to that line only when

x\ne-2,

so it omits

(-2,-4).

In equation III, every point with

x=-2

satisfies

0=0;

for other

x

it gives

y=x-2.

Thus III is the line together with the entire vertical line

x=-2.

The three graphs are different.

Thus the correct answer is E.

Problem 14 in other years