document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=698014>Question 698014</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I am so stumped on this I have two problems actually on my homework I have no idea where to start or even know how to attempt...\r<BR>\n" );
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document.write( "1. Given the following quadratic equation, find a. the vertex b. the axis of symmetry c. the intercepts<BR>\n" );
document.write( "d. the domain e. the range f. the interval where the function is increasing, and g. the interval where the function is decreasing h. Graph the function.<BR>\n" );
document.write( "y=2x^2+12x+20\r<BR>\n" );
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document.write( "2.&#65279;Given the following polynominal, find<BR>\n" );
document.write( "a. the zeros and the multiplicity of each<BR>\n" );
document.write( "b.where the graph crosses or touches the x-axis<BR>\n" );
document.write( "c. number of turning points<BR>\n" );
document.write( "d. end behavior\r<BR>\n" );
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document.write( "f(x)=-x^2(x+6)^3(x^2-1)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #430411 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=430411\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "For any quadratic of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \rho(x)\ =\ ax^2\ +\ bx\ +\ c\">, the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">-coordinate of the vertex is found by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x_v = \frac{-b}{2a}\">\r<BR>\n" );
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document.write( "and the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-coordinate of the vertex is found by:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_v = \rho(x_v) = \rho\left(\frac{-b}{2a}\right)\ =\ a\left(\frac{-b}{2a}\right)^2\ +\ b\left(\frac{-b}{2a}\right)\ +\ c\">\r<BR>\n" );
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document.write( "An equation of the axis of symmetry is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ x\ =\ x_v\">\r<BR>\n" );
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document.write( "which is to say the vertical line that passes through the vertex.\r<BR>\n" );
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document.write( "To find the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-intercept, set <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\ =\ 0\">, then calculate the resulting value of the function, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\ =\ \beta\">.  Then the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-intercept is the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE (0,\beta)\">\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">-intercepts are the points <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\alpha_1,0\right)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(\alpha_2,0\right)\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha_1\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \alpha_2\"> are the real number zeros of the function.  The first thing to do is to determine whether the function has any real zeros.\r<BR>\n" );
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document.write( "<u>Nature of zeroes</u>\r<BR>\n" );
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document.write( "Find the Discriminant, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta\ =\ b^2\ -\ 4ac\"> and evaluate the nature of the zeroes as follows:\r<BR>\n" );
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document.write( "No calculation quick look:  If the signs on <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large a\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large c\"> are opposite, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta > 0\"> guaranteed.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta > 0 \ \ \Rightarrow\ \\"> Two real and unequal zeroes. If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta\"> is a perfect square, the quadratic factors over <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \mathbb{Q}\"> (the rationals).\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta = 0 \ \ \Rightarrow\ \\"> One real zero with a multiplicity of two.  That is to say that the trinomial is a perfect square and has two identical factors.  Presuming rational coefficients, the zero will be rational as well.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \Delta < 0 \ \ \Rightarrow\ \\"> A conjugate pair of complex number zeroes of the form <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a \pm bi\"> where <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE i\"> is the imaginary number defined by <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE i^2 = -1\">.  In this case, a quadratic function does NOT intercept the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">-axis.\r<BR>\n" );
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document.write( "To find the domain, determine the set of real numbers for which the function is defined.  Hint:  The domain of ALL polynomial functions is the set of real numbers.  In interval notation, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \left(-\infty,\infty\right)\">.  In set notation:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{dom}\{\rho(x)\}\ =\ \{x\,\in\,\mathbb{R}\}\">\r<BR>\n" );
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document.write( "To find the range, determine whether the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-coordinate  is of the vertex is a minimum or a maximum value of the function.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\ >\ 0\">, then the graph of the quadratic is a parabola that opens upward, meaning that the vertex is at the bottom of the graph, so the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-coordinate of the vertex is a ______________.  (You fill in the blank).  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a\ <\ 0\">, then we have the opposite case.  If the vertex represents a minimum, then the lower limit of the range is the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-coordinate of the vertex and the upper limit increases without bound. If the vertex represents a maximum, then the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-coordinate represents the upper limit of the range and the lower limit decreases without bound.  The other Hence, the range is represented either as an interval or a set.\r<BR>\n" );
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document.write( "Interval Notation:  (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?-\infty,y_v\">] or [<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?y_v,\infty\">)\r<BR>\n" );
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document.write( "Sets:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{rng}\{\rho(x)\}\ =\ \{y,\in\,\mathbb{R}\,=\,\rho(x)\ |\ y_v\ \leq\ y\ <\ \infty\}\">\r<BR>\n" );
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document.write( "or\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \text{rng}\{\rho(x)\}\ =\ \{y,\in\,\mathbb{R}\,=\,\rho(x)\ |\ -\infty\ <\ y\ \leq\ y_v\}\">\r<BR>\n" );
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document.write( "Regions of increase or decrease.  Start with a value for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> that is strictly less than <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x_v\">.  Determine the value of the function at this selected <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> value.  Compare the calculated function value with <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y_v\">.  If the calculated function value is LARGER than <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y_v\">, then the function is DECREASING on the interval (<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?-\infty,x_v\">].  In this case, the function is, perforce, increasing on the other half-plane interval, to wit: [<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?x_v,\infty\">).  On the other hand, if the calculated function value when the selected value of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> is SMALLER than <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y_v\">, then the function is INCREASING on the side of the graph to the left of the axis of symmetry and decreasing on the other side.\r<BR>\n" );
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document.write( "<u>Graphing</u>\r<BR>\n" );
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document.write( "1.  Plot the vertex.\r<BR>\n" );
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document.write( "2.  Plot the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-intercept\r<BR>\n" );
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document.write( "3.  Plot the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">-intercepts if they exist\r<BR>\n" );
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document.write( "4.  Use symmetry to find a point that corresponds to the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-intercept.  Determine the distance along the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">-axis between the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-axis and the axis of symmetry.  Move that same distance on the OTHER side of the axis of symmetry.  At that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\">-value, there will be a point that has the same function value (that is the same <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-value] as the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-value of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">-intercept.  In general, points on the graph of the parabola equidistant from the axis of symmetry have identical function (or <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y\">) values.\r<BR>\n" );
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document.write( "5.  Pick another <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x\"> value that has not been used.  Calculate the function value and plot the point.  Use symmetry to find the corresponding point on the other side of the axis of symmetry.\r<BR>\n" );
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document.write( "6.  When you have enough points plotted to suit yourself, draw a smooth curve through the plotted points.\r<BR>\n" );
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document.write( "<u>The importance of reading instructions</u>.  One of the reasons you may be having difficulty with mathematics is your demonstrated inability to either read or follow (or both) written instructions.  Before you post in the future, please take some time to read and understand the instructions for posting on this site.\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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