document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=638927>Question 638927</A>: <I><SPAN STYLE=\"word-break: break-all;\"> determine the quadratic function f whose vertex is (2,-4) and passes through (1,-2)\r<BR>\n" );
document.write( "\n" );
document.write( "f(x)=?<BR>\n" );
document.write( "answer question in form of f(x)=ax^2+bx+c </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #402490 by </B><A HREF=/tutors/aboutme.mpl?userid=DrBeeee><B>DrBeeee(684)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=DrBeeee><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=402490\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> General equation of a quadratic is <BR>\n" );
document.write( "(1) f(x) = ax^2 + bx +c.<BR>\n" );
document.write( "The graph of this quadratic is called a parabola. It is a curve shaped like the letter \"U\" or an upside down \"U\" depending on the sign of \"a\", the coefficient of x^2. If a>0 the parabola opens \"up\", like U. If a<0, then parabola opens \"down\". A useful formula is the value of x around which the parabola is symmetrical. This value is given by<BR>\n" );
document.write( "(2) x = -b/(2a)<BR>\n" );
document.write( "The value of x in (2) is also value of the x coordinate of the parabola's vertex, the point where the graph of f(x) hits a maximum or minimum.<BR>\n" );
document.write( "For this problem we are given two points through which the parabola passes. The point <BR>\n" );
document.write( "(3) (1,-2)<BR>\n" );
document.write( "and and the vertex<BR>\n" );
document.write( "(4) (2,-4).<BR>\n" );
document.write( "Using (2) and the value of x in the vertex (4) yields<BR>\n" );
document.write( "2 = -b/(2a) or<BR>\n" );
document.write( "(5) b = -4a<BR>\n" );
document.write( "Substituting b into (1) yields<BR>\n" );
document.write( "(6) y = ax^2 -4ax + c<BR>\n" );
document.write( "Now substitute the point (1,-2) of (3) into (6) yields<BR>\n" );
document.write( "(7) -2 = a*(1)^2 - 4a*(1) + c, or simplifying<BR>\n" );
document.write( "(8) c = -3a - 2<BR>\n" );
document.write( "Now substitute the vertex (2,-4) of (4) into (6) yields<BR>\n" );
document.write( "(9) -4 = a*(2)^2 -4a*(2) + c, or simplifying<BR>\n" );
document.write( "(10) c = 4a - 4<BR>\n" );
document.write( "Equating (8) and (10), and simplifying yields<BR>\n" );
document.write( "(11) a = 2<BR>\n" );
document.write( "Put a = 2 into (5) yields<BR>\n" );
document.write( "(12) b = -8<BR>\n" );
document.write( "Put a = 2 into (8) and simplifying yields<BR>\n" );
document.write( "(13) c = 4<BR>\n" );
document.write( "Check our answer in (7)<BR>\n" );
document.write( "Is [-2 = 2*1 - 4*2*1 + 4]?<BR>\n" );
document.write( "Is [-2 = 2 - 8 + 4]?<BR>\n" );
document.write( "Is [-2 = -2]? Yes<BR>\n" );
document.write( "Check our answer in (9)<BR>\n" );
document.write( "Is [-4 = 2*2^2 -4*2*2 +4]?<BR>\n" );
document.write( "Is [-4 = 8 - 16 +4]?<BR>\n" );
document.write( "Is [-4 = -4]? Yes<BR>\n" );
document.write( "Answer: f(x) = 2x^2 -8x +4 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );