document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=963469>Question 963469</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Part I. <BR>\n" );
document.write( "G(x) = -x4 + 32x2 +144 <BR>\n" );
document.write( "a.  Show whether G is an even, odd function or neither. If it’s even/odd, identify the property of its graph. (Must show your work)<BR>\n" );
document.write( "(Help: To show that G is an even function, you must show that G(-x) = G(x), and G is an odd function you must show that G(-x) = -G(x), for ex. F(x) = x3 – 3x is an odd function, since:<BR>\n" );
document.write( " F(-x) = (-x)3-3(-x) = -x3+3x = -(x3-3x) = -F(x), thus F is an odd function. And by definition odd function has graph symmetric about the origin).\r<BR>\n" );
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document.write( "b. There is a local maximum value of 400 at x = 4, determine the second local maximum value (Use property of the graph of an even/odd function from part a:<BR>\n" );
document.write( " (Hints:  if (x, f(x)) is a point on the graph of an even function f, then (-x, f(-x)) is symmetric  about the y-axis since f(-x)= f(x) is also on the graph of f).\r<BR>\n" );
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document.write( "c. Suppose the area under the graph of G between x = 0, and x = 6, that is bounded below by the x-axis is 1612.8 square units, Using the result from part a to determine the area under the graph of G between x = -6 and x = 0, bounded below by the x-axis.<BR>\n" );
document.write( " (Hints: Use property of the graph of an even/odd function).\r<BR>\n" );
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document.write( "Part II. <BR>\n" );
document.write( "a. Choose a library function, then use the library function, and describe how to graph the function:<BR>\n" );
document.write( "Y = (x – 3)2 – 5 (just describe without graphing).<BR>\n" );
document.write( "(Hints: Y = |x + 4| + 7, then library function is f(x) = |x| absolute value function, then the graph of Y can be obtained from the graph of f by shifting the graph of f to the left 4 units, and up 7 units).\r<BR>\n" );
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document.write( "b. Use the library function f(x) =&#8730;x Write the function y obtained from f(x) =&#8730;x by shifting the graph of f 4 units to the left, and down 2 unit.<BR>\n" );
document.write( "(Hints: f(x) = x2; f(x) = x3; f(x) = &#8730;x; f(x) = |x|,  etc.. each is a library function. If y= f(x-h), then graph of y can be obtained from graph of f by shifting the graph of f to the right h units, etc…Please review transformations  of graphs = shifting technique to answer both questions of part a, and b).\r<BR>\n" );
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document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #588629 by </B><A HREF=/tutors/aboutme.mpl?userid=MathLover1><B>MathLover1(20849)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=MathLover1><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/cgi-bin/show-face.mpl?user=MathLover1><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=588629\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Part I. \r<BR>\n" );
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document.write( "a.\r<BR>\n" );
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document.write( "A function is \"even\" when:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+f%28-x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=+f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "A function is \"odd\" when:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-f%28x%29+=+f%28-x%29 ALIGN=MIDDLE  ALT=\"-f%28x%29+=+f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "check your function:  is it \"even\"<BR>\n" );
document.write( "f(x) = f(-x) for all x<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28-x%29++=++-%28-x%29%5E4%2B32%28-x%29%5E2%2B144 ALIGN=MIDDLE  ALT=\"G%28-x%29++=++-%28-x%29%5E4%2B32%28-x%29%5E2%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" >...since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%29%5E4 ALIGN=MIDDLE  ALT=\"%28-x%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be same as<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28x%29%5E4 ALIGN=MIDDLE  ALT=\"+%28x%29%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-x%29%5E2 ALIGN=MIDDLE  ALT=\"%28-x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be same as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%29%5E2 ALIGN=MIDDLE  ALT=\"%28x%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we have\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28-x%29++=++-x%5E4%2B32x%5E2%2B144 ALIGN=MIDDLE  ALT=\"G%28-x%29++=++-x%5E4%2B32x%5E2%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "so, your function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=even ALIGN=MIDDLE  ALT=\"even\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "is it odd:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-f%28x%29+=+f%28-x%29 ALIGN=MIDDLE  ALT=\"-f%28x%29+=+f%28-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > for all <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+-%28+-x%5E4%2B32x%5E2%2B144%29=-%28-x%29%5E4%2B32%28-x%29%5E2%2B144 ALIGN=MIDDLE  ALT=\"+-%28+-x%5E4%2B32x%5E2%2B144%29=-%28-x%29%5E4%2B32%28-x%29%5E2%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+x%5E4-32x%5E2-144%3C%3E-x%5E4%2B32x%5E2%2B144 ALIGN=MIDDLE  ALT=\"+x%5E4-32x%5E2-144%3C%3E-x%5E4%2B32x%5E2%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" >.......your function is not <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=odd ALIGN=MIDDLE  ALT=\"odd\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "b.<BR>\n" );
document.write( "There is a local maximum value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400 ALIGN=MIDDLE  ALT=\"400\"   DISABLED_style= \"max-width:90%; height:auto;\" > at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4 ALIGN=MIDDLE  ALT=\"x+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, determine the second local maximum value:\r<BR>\n" );
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document.write( "(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x ALIGN=MIDDLE  ALT=\"x\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+f%28x%29 ALIGN=MIDDLE  ALT=\"+f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) =(<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400 ALIGN=MIDDLE  ALT=\"400\"   DISABLED_style= \"max-width:90%; height:auto;\" >)<BR>\n" );
document.write( "find<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28-4%29++=++-%28-4%29%5E4%2B32%28-4%29%5E2%2B144 ALIGN=MIDDLE  ALT=\"G%28-4%29++=++-%28-4%29%5E4%2B32%28-4%29%5E2%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28-4%29++=++-256%2B32%2A16%2B144 ALIGN=MIDDLE  ALT=\"G%28-4%29++=++-256%2B32%2A16%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28-4%29++=++-256%2B512%2B144 ALIGN=MIDDLE  ALT=\"G%28-4%29++=++-256%2B512%2B144\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28-4%29++=+400 ALIGN=MIDDLE  ALT=\"G%28-4%29++=+400\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " so, if (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400 ALIGN=MIDDLE  ALT=\"400\"   DISABLED_style= \"max-width:90%; height:auto;\" >) is a point on the graph of an even function G, then (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-4 ALIGN=MIDDLE  ALT=\"-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=400 ALIGN=MIDDLE  ALT=\"400\"   DISABLED_style= \"max-width:90%; height:auto;\" >)  is symmetric about the y-axis since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=G%28-x%29=+G%28x%29 ALIGN=MIDDLE  ALT=\"G%28-x%29=+G%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is also on the graph of G\r<BR>\n" );
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document.write( "c. Suppose the area under the graph of G between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+6 ALIGN=MIDDLE  ALT=\"x+=+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >, that is bounded below by the x-axis is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1612.8 ALIGN=MIDDLE  ALT=\"1612.8\"   DISABLED_style= \"max-width:90%; height:auto;\" > square units, Using the result from part a to determine the area under the graph of G between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-6 ALIGN=MIDDLE  ALT=\"x+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, bounded below by the x-axis.\r<BR>\n" );
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document.write( "Graphs of even and odd functions have following properties:\r<BR>\n" );
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document.write( "    If function is even then its graph is symmetric about y-axis.<BR>\n" );
document.write( "    If function is odd then its graph is symmetric about origin.\r<BR>\n" );
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document.write( "so, because the  graph is symmetric about y-axis,  the area under the graph of G between <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+-6 ALIGN=MIDDLE  ALT=\"x+=+-6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+0 ALIGN=MIDDLE  ALT=\"x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >  is bounded below by the x-axis is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1612.8 ALIGN=MIDDLE  ALT=\"1612.8\"   DISABLED_style= \"max-width:90%; height:auto;\" > square units\r<BR>\n" );
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document.write( "Part II. \r<BR>\n" );
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document.write( "a.\r<BR>\n" );
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document.write( "Choose a library function, then use the library function, and describe how to graph the function:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y+=+%28x-3%29%5E2+-5 ALIGN=MIDDLE  ALT=\"Y+=+%28x-3%29%5E2+-5\"   DISABLED_style= \"max-width:90%; height:auto;\" > (just describe without graphing).\r<BR>\n" );
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document.write( "library function is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=x%5E2 ALIGN=MIDDLE  ALT=\"f%28x%29+=x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=Y ALIGN=MIDDLE  ALT=\"Y\"   DISABLED_style= \"max-width:90%; height:auto;\" > can be obtained from the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by shifting the graph of f to the right <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3 ALIGN=MIDDLE  ALT=\"3\"   DISABLED_style= \"max-width:90%; height:auto;\" > units, and down <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5 ALIGN=MIDDLE  ALT=\"5\"   DISABLED_style= \"max-width:90%; height:auto;\" > units\r<BR>\n" );
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document.write( " b.\r<BR>\n" );
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document.write( " Use the library function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=sqrt%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Write the function y obtained from <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=sqrt%28x%29 ALIGN=MIDDLE  ALT=\"f%28x%29+=sqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by shifting the graph of f <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4 ALIGN=MIDDLE  ALT=\"4\"   DISABLED_style= \"max-width:90%; height:auto;\" > units to the left, and down <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2 ALIGN=MIDDLE  ALT=\"2\"   DISABLED_style= \"max-width:90%; height:auto;\" > unit.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=sqrt%28x%2B4%29+-2 ALIGN=MIDDLE  ALT=\"f%28x%29+=sqrt%28x%2B4%29+-2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+graph%28+600%2C+600%2C+-10%2C+10%2C+-10%2C+10%2C-sqrt%28x%29%2C+-sqrt%28x%2B4%29+-2%2C+sqrt%28x%29%2C+sqrt%28x%2B4%29+-2%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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