document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199626>Question 1199626</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How does the graph of the hyperbola whose equation is (y2)/4 - (x2)/9 = 1 open?\r<BR>\n" );
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document.write( "A. A hyperbola does not open.<BR>\n" );
document.write( "B. Hyperbola opens to the sides.<BR>\n" );
document.write( "C. Hyperbola opens toward its vertices.<BR>\n" );
document.write( "D. This graph is not a hyperbola.<BR>\n" );
document.write( "E. Hyperbola opens toward its center.<BR>\n" );
document.write( "F. Hyperbola opens up and down. </SPAN>\n" );
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document.write( "Note for future reference: It is standard to use \"^\" (shift-6) to represent exponentiation.  The equation in your problem is (y^2)/4 - (x^2)/9 = 1.<br><BR>\n" );
document.write( "The equation has both x^2 and y^2 terms, with opposite signs, and it contains no xy term.  So the graph is a hyperbola that opens either right and left (option B -- \"to the sides\") or up and down (option F).<br><BR>\n" );
document.write( "We need to determine which of those options is the right one.<br><BR>\n" );
document.write( "In my experience, a lot of references teach a rule: if the x^2 term is positive, the hyperbola opens in the x direction (right and left); if the y^2 term is positive, the hyperbola opens in the y direction (up and down).<br><BR>\n" );
document.write( "That works... but if a student just memorizes a rule and then forgets the rule or gets confused about what the rule says, then he will have trouble finding the answer.<br><BR>\n" );
document.write( "I recommend using a method that uses logical reasoning rather than memorization.<br><BR>\n" );
document.write( "What I do when I see your equation is suppose that y=0; then the equation says <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-x%5E2%2F9=1 ALIGN=MIDDLE  ALT=\"-x%5E2%2F9=1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which makes x^2 negative, which is impossible.  That means y can't be zero -- and that means the hyperbola opens up and down.<br><BR>\n" );
document.write( "Using logical reasoning rather than a memorized rule helps the student UNDERSTAND why the graph opens in the direction it does; and that makes the student enjoy the math more.<br><BR>\n" );
document.write( "ANSWER: up and down -- option F<br><BR>\n" );
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