document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=199593>Question 199593</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi all, I was hoping someone could help me with the following Graphing problems im having. I need to find the x and y intercepts, if any, and turning points for the following functions. (Each one on seperate graphs)<BR>\n" );
document.write( "1. y = 2x^2-8x+9<BR>\n" );
document.write( "2. y = 2x^2-3x-1<BR>\n" );
document.write( "3. y = -2x^2-7x+4<BR>\n" );
document.write( "4. y = x^2-x-2\r<BR>\n" );
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document.write( "I f you could please show the working with descriptions on each equations that would be very helpful. A fully labelled sketch of the function including the points and axis of symmetry would also be great.<BR>\n" );
document.write( "Thanks, -Nick </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #150019 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=150019\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Garamond\" size=\"+2\">\r<BR>\n" );
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document.write( "It seems like you want someone to do all of your work for you.  Not going to happen.  What I will do is show you the <i>process</i> for doing all of them, so that you can do your own work.\r<BR>\n" );
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document.write( "First thing is that these are all quadratic functions of the form:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y = f(x) = ax^2 + bx + c\">\r<BR>\n" );
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document.write( "And all subsequent discussion will be related to the components of this form.\r<BR>\n" );
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document.write( "<u><i><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-intercept</i></u>\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-intercept is the point where the graph intersects the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-axis.  The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of this point is obviously 0, and the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate is the value of the function when <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = 0\">, specifically:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y = f(0) = a(0)^2 + b(0) + c = c\">\r<BR>\n" );
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document.write( "Therefore 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 \left(0,c\right)\"> \r<BR>\n" );
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document.write( "<u><i><IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercept(s)</i></u>\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercept(s) are the point(s) where the graph intersects the  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis.  The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinates of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts correspond to the real zeros of the function; in other words the real number solutions or roots of the equation:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ ax^2 + bx + c = 0\">\r<BR>\n" );
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document.write( "Recall the discriminant for a quadratic equation.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta = b^2 - 4ac > 0\">, you have two <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts.  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta = 0\">, you have one <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercept (and the graph is tangent to the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-axis at the vertex of the parabola).  If <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \Delta < 0\"> you have zero <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercepts.\r<BR>\n" );
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document.write( "The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of an <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-intercept is 0.  To determine the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinates, set the function equal to 0 and solve for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> using any appropriate method (factoring, completing the square, or the quadratic formula).\r<BR>\n" );
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document.write( "<u><i>Turning Points</i></u>\r<BR>\n" );
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document.write( "Since these are all quadratic functions of the form defined above, all of the graphs are of parabolas.  Therefore, there is one and only one turning point, and that is the vertex of the parabola.  The <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex of the graph of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y = f(x) = ax^2 + bx + c\"> is given 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 the value of the function at the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\">-coordinate of the vertex, i.e:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ y_v = f(x_v) = f\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( "<u><i>Axis of Symmetry</i></u>\r<BR>\n" );
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document.write( "The axis of symmetry for any function in this form is the vertical line described by the equation: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = x_v\">, that is to say <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = \frac{-b}{2a}\">\r<BR>\n" );
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document.write( "<u><i>Additional points</i></u>\r<BR>\n" );
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document.write( "If you feel the need to determine additional points to help you sketch the true shape of the graph, then just select values for <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x\"> different than the values that you calculated above.  Because of symmetry, the graph will have a point at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x = \frac{-b}{a}\"> that has a <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate equal to the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate of the <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-intercept.  Also because of symmetry, if the point <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \left(x_1,y_1\right)\"> is a point on the graph, and if <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_1 = \frac{-b}{2a} + p\">, then the point at <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large x_2 = \frac{-b}{2a} - p\"> will also have a function value (read <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y\">-coordinate) of <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large y_1\">\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" );
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