document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=166503>Question 166503</A>: <I><SPAN STYLE=\"word-break: break-all;\"> PLEASE HELP!!!<BR>\n" );
document.write( "Graph f(x)=-x^2+4x-3, labeling the y intercept, vertex, and axis of symmetry.<BR>\n" );
document.write( "Also <BR>\n" );
document.write( "graph f(x)=(-x)^4+4x^2-2x by making a table of values.  Then estimate the x-coordinates at which the relative maxima and relative minima occur.<BR>\n" );
document.write( "I must show my work.  This is a worksheet.<BR>\n" );
document.write( "Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #122711 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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=122711\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> first part:<BR>\n" );
document.write( "Graph f(x)=-x^2+4x-3, labeling the y intercept, vertex, and axis of symmetry.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "equation is f(x) = -x^2 + 4x -3<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since there is a minus sign in front of the x^2, this equation will point up and open down.<BR>\n" );
document.write( "this means that the vertex will be a maximum point.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since this equation is in the form of:<BR>\n" );
document.write( "a*x^2 + b*x + c,<BR>\n" );
document.write( "the formula for the maximum / minimum point of the graph is:<BR>\n" );
document.write( "x = -b/2a<BR>\n" );
document.write( "this point will be a maximum point.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "b = 4<BR>\n" );
document.write( "a = -1<BR>\n" );
document.write( "-b / 2a = (-4)/(-2) = 2<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the x value of the maximum point of the graph is:<BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "the y value of the maximum point of the graph is found by substituting x = 2 in the equation.<BR>\n" );
document.write( "the equation is:<BR>\n" );
document.write( "f(x) = -x^2 + 4x -3<BR>\n" );
document.write( "substituting 2 for x makes the equation become:<BR>\n" );
document.write( "f(2) = -(2)^2 + 4(2) - 3<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "- 4 + 8 - 3 = 1 <BR>\n" );
document.write( "the y value of the maximum point is 1.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the coordinates of the maximum point are:<BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "y = 1<BR>\n" );
document.write( "coordinates (x,y) are:  (2,1)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the vertex is the maximum point of this graph.<BR>\n" );
document.write( "the x value of the vertex is the line of symmetry.<BR>\n" );
document.write( "line of symmetry = x = 2<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the y intercept of this graph is found by making x = 0<BR>\n" );
document.write( "f(0) = -(0)^2 + 4*0 - 3<BR>\n" );
document.write( "f(0) = -3<BR>\n" );
document.write( "y intercept of this graph is -3.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the graph looks like this:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C800%2C-5%2C5%2C-5%2C5%2C-x%5E2+%2B+4x+-+3%29 ALIGN=MIDDLE  ALT=\"graph%28800%2C800%2C-5%2C5%2C-5%2C5%2C-x%5E2+%2B+4x+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "as you can see, the y intercept is -3 (value of y when x = 0).<BR>\n" );
document.write( "the vertex is the maximum point of the graph = (2,1)<BR>\n" );
document.write( "the axis of symmetry = x = 2.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "second part:<BR>\n" );
document.write( "graph f(x)=(-x)^4+4x^2-2x by making a table of values.  Then estimate the x-coordinates at which the relative maxima and relative minima occur.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this graph looks like it will open upwards and point downward since all values of f(x) look like they will be positive.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "to prove this: plot of 7 points should be more than sufficient.<BR>\n" );
document.write( "the points to be plotted are<BR>\n" );
document.write( "(x,y)<BR>\n" );
document.write( "(-3,123)<BR>\n" );
document.write( "(-2,36)<BR>\n" );
document.write( "(-1,7)<BR>\n" );
document.write( "(0,0)<BR>\n" );
document.write( "(1,3)<BR>\n" );
document.write( "(2,28)<BR>\n" );
document.write( "(3,111)\r<BR>\n" );
document.write( "\n" );
document.write( "-----<BR>\n" );
document.write( "the x coordinates of the relative minima are 0.<BR>\n" );
document.write( "there are no maxima.<BR>\n" );
document.write( "graph will open upwards and continue to infinity.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "graph of this equation looks like:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28800%2C800%2C-5%2C5%2C-10%2C200%2C%28-x%29%5E4%2B4x%5E2-2x%29 ALIGN=MIDDLE  ALT=\"graph+%28800%2C800%2C-5%2C5%2C-10%2C200%2C%28-x%29%5E4%2B4x%5E2-2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );