document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1162807>Question 1162807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1) Lets look at some examples of different functions. Each of the following functions has a zero located at x = 3. For each function, determine if the function’s graph crosses or only touches the x-axis at x = 3.\r<BR>\n" );
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document.write( "f(x)=x(x-3)^2<BR>\n" );
document.write( "g(x) = x(x+2)(x-3)<BR>\n" );
document.write( "h(x)=(x-3)^3<BR>\n" );
document.write( "2) . Do some additional examples for yourself using functions you create yourself by multiplying two or three linear factors so that there are clear zeros. Feel free to use squared or cubed linear factors like f(x) and h(x) above. Is there a pattern to when the graph of the function only touches the x-axis and when it actually crosses it?\r<BR>\n" );
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document.write( "3)The degree of a polynomial can be found by fully expanding the polynomial. Once this is done, look at the variable in the polynomial with the largest exponent. This exponent is the degree of the polynomial. The leading coefficient of a polynomial is the coefficient of the term with the largest exponent. Look at the graphs of each of the examples you have worked with so far and expand them using multiplication. Do you notice a pattern of when a graph starts below or above the x-axis? Do you notice a pattern when a graph ends below or above the x-axis?\r<BR>\n" );
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document.write( "4)Create a function that does the following: • It touches the x-axis at two points, x = -3 and x = 1 • It begins above the x-axis • It ends below the x-axis. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #786710 by </B><A HREF=/tutors/aboutme.mpl?userid=Boreal><B>Boreal(15235)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Boreal><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=Boreal><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=786710\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> f(x)=x(x-3)^2 roots at 0 and 3, with a bounce at 3.  Crosses and touches<BR>\n" );
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document.write( "g(x)=x(x+2)(x-3), roots at 0,-2, 3 crosses<BR>\n" );
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document.write( "H(x)=(x-3)^3 crosses at x=3<BR>\n" );
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document.write( "4. -(x+3)(x-1)^2. This is a cubic, so making it negative will have the beginning positive and the end negative. Put in a bounce as the last pair of roots, and the function crosses once and touches.<BR>\n" );
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