document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190934>Question 1190934</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The tangent to the curve y = 2x^2 + ax + b at the point (-2,11) is perpendicular to the line 2y = x + 7. Find the value of a and b. </SPAN>\n" );
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document.write( "We'll need the derivative.<BR>\n" );
document.write( "y = 2x^2 + ax + b<BR>\n" );
document.write( "dy/dx = 4x + a\r<BR>\n" );
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document.write( "Plug in the x coordinate of (-2,11)<BR>\n" );
document.write( "dy/dx = 4x + a<BR>\n" );
document.write( "dy/dx = 4(-2) + a<BR>\n" );
document.write( "dy/dx = -8 + a<BR>\n" );
document.write( "This is the slope of the tangent line at the point (-2,11)\r<BR>\n" );
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document.write( "Let's solve the other equation for y<BR>\n" );
document.write( "2y = x+7<BR>\n" );
document.write( "y = x/2+7/2<BR>\n" );
document.write( "y = 0.5x+3.5<BR>\n" );
document.write( "The slope is 1/2 = 0.5<BR>\n" );
document.write( "The tangent slope we found earlier is perpendicular to this. The negative reciprocal of 1/2 is -2/1 or just -2.\r<BR>\n" );
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document.write( "Set the slope of -8+a equal to the -2 and solve for 'a'<BR>\n" );
document.write( "-8+a = -2<BR>\n" );
document.write( "a = -2+8<BR>\n" );
document.write( "a = 6\r<BR>\n" );
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document.write( "We get<BR>\n" );
document.write( "dy/dx = -8+a<BR>\n" );
document.write( "dy/dx = -8+6<BR>\n" );
document.write( "dy/dx = -2<BR>\n" );
document.write( "which helps confirm things so far.\r<BR>\n" );
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document.write( "The original equation<BR>\n" );
document.write( "y = 2x^2 + ax + b<BR>\n" );
document.write( "updates to<BR>\n" );
document.write( "y = 2x^2 + 6x + b\r<BR>\n" );
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document.write( "Now plug in (x,y) = (-2,11) and solve for b<BR>\n" );
document.write( "y = 2x^2 + 6x + b<BR>\n" );
document.write( "11 = 2(-2)^2 + 6(-2) + b<BR>\n" );
document.write( "11 = -4 + b<BR>\n" );
document.write( "b = 11+4<BR>\n" );
document.write( "b = 15\r<BR>\n" );
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document.write( "Visual confirmation using a graph<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"https://i.imgur.com/5pLpFt3.png\"><BR>\n" );
document.write( "Graph was made with GeoGebra.\r<BR>\n" );
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document.write( "Answers:<BR>\n" );
document.write( "a = 6<BR>\n" );
document.write( "b = 15<BR>\n" );
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