document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209789>Question 1209789</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Draw the graph of y=x²+x+2<BR>\n" );
document.write( " For -4≤x≤4<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 #850429 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850429\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> To draw the graph of $y = x^2 + x + 2$, we need to determine its key features:\r<BR>\n" );
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document.write( "1. **Vertex:** The vertex of the parabola is at the point $x = \dfrac{-b}{2a} = \dfrac{-1}{2(1)} = -\dfrac{1}{2}$.  Substituting this value of $x$ into the equation, we get $y = \left( -\dfrac{1}{2} \right)^2 - \dfrac{1}{2} + 2 = \dfrac{7}{4}$.  So, the vertex is at $\left( -\dfrac{1}{2}, \dfrac{7}{4} \right)$.\r<BR>\n" );
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document.write( "2. **Axis of Symmetry:** The axis of symmetry is the vertical line passing through the vertex, which is $x = -\dfrac{1}{2}$.\r<BR>\n" );
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document.write( "3. **y-intercept:** The y-intercept is the point where the graph intersects the y-axis.  Setting $x = 0$, we get $y = 2$.  So, the y-intercept is at $(0,2)$.\r<BR>\n" );
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document.write( "4. **x-intercepts:** The x-intercepts are the points where the graph intersects the x-axis.  Setting $y = 0$, we get the quadratic equation $x^2 + x + 2 = 0$.  The discriminant of this quadratic is $1^2 - 4 \cdot 1 \cdot 2 = -7$, which is negative, so the quadratic has no real roots.  This means the graph does not intersect the x-axis.\r<BR>\n" );
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document.write( "5. **Direction:** Since the coefficient of $x^2$ is positive, the parabola opens upward.\r<BR>\n" );
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document.write( "Now, we can plot the graph:\r<BR>\n" );
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document.write( "[asy]<BR>\n" );
document.write( "unitsize(2 cm);\r<BR>\n" );
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document.write( "real func (real x) {<BR>\n" );
document.write( "  return(x^2 + x + 2);<BR>\n" );
document.write( "}\r<BR>\n" );
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document.write( "draw(graph(func,-4,4));<BR>\n" );
document.write( "draw((-4,0)--(4,0));<BR>\n" );
document.write( "draw((0,-1)--(0,18));<BR>\n" );
document.write( "label(\"$y = x^2 + x + 2$\", (4,func(4)), E);\r<BR>\n" );
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document.write( "dot(\"$\left( -\frac{1}{2}, \frac{7}{4} \right)$\", (-1/2,7/4), S);<BR>\n" );
document.write( "dot(\"$y$-intercept\", (0,2), NE);<BR>\n" );
document.write( "[/asy] </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );