document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=346043>Question 346043</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Graph f(x)=-x^2+4x-3, labeling the y-intercept, vertex, and axis of symmetry </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #247439 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=247439\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28600%2C600%2C-5%2C10%2C-16%2C16%2C-x%5E2+%2B+4x+-+3%29 ALIGN=MIDDLE  ALT=\"graph%28600%2C600%2C-5%2C10%2C-16%2C16%2C-x%5E2+%2B+4x+-+3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "looks like your y-intercepts is at (0,-3).\r<BR>\n" );
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document.write( "looks like your x-intercepts are at (1,0) and (3,0).\r<BR>\n" );
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document.write( "looks like  your axis of symmetry is at x = 2.\r<BR>\n" );
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document.write( "looks like your vertex is at (2,1).\r<BR>\n" );
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document.write( "coordinates are shown as (x,y) where x is the value along the x-axis and y is the value along the y-axis.\r<BR>\n" );
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document.write( "the point (2,1) is the intersection of a vertical line at x = 2 and a horizontal line at y = 1.\r<BR>\n" );
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document.write( "you find the x-intercepts by solving for the roots of the equation.\r<BR>\n" );
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document.write( "your equation is -x^2 + 4x - 3.\r<BR>\n" );
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document.write( "to find the roots you set the equation equal to 0.\r<BR>\n" );
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document.write( "you get -x^2 + 4x - 3 = 0\r<BR>\n" );
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document.write( "it's easier to factor if the coefficient of the x^2 term is positive so multiply both sides of the equation by -1 to get:\r<BR>\n" );
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document.write( "x^2 - 4x + 3 = 0\r<BR>\n" );
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document.write( "multiplying both sides of an equality by -1 keeps the equality so you are changing the equation, but not changing the equal relations between the 2 sides of the equation.\r<BR>\n" );
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document.write( "the equal relationship is what's important to solving for the roots, so changing the equation by multiplying both sides by -1 will give you the same answer.\r<BR>\n" );
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document.write( "down below, at the end, i'll show you how the factors worked out if you did not multiply both sides of the eqution by -1 before factoring.\r<BR>\n" );
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document.write( "your factors appear to be (x-3) * (x-1) = 0\r<BR>\n" );
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document.write( "if you multiply (x-3) * (x-1), you will get x^2 -x -3x + 3 which simplifies to x^2 - 4x + 3.\r<BR>\n" );
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document.write( "this makes the value of y equal to 0 then x = 3 or x = 1.\r<BR>\n" );
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document.write( "plug 3 into your original equation and you will see that it will equal to 0.\r<BR>\n" );
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document.write( "plus 1 into your original equation and you will see that it will also equal to 0.\r<BR>\n" );
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document.write( "the graph confirms that the values of x are 1 and 3 when the value of y is equal to 0.\r<BR>\n" );
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document.write( "since the coefficient of your x^2 term in the original equation is negative, the graph of the equation will point up and open down.\r<BR>\n" );
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document.write( "that means that the vertex of the equation will be a maximum point.\r<BR>\n" );
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document.write( "if the coefficient of your x^2 term was positive, the graph ofr the equation would have pointed down and open up.   \r<BR>\n" );
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document.write( "that means that the vertex of the equation would have been a minimum point.\r<BR>\n" );
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document.write( "but it's not.\r<BR>\n" );
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document.write( "it's a maximum point because the coefficient of the x^2 term is negative.\r<BR>\n" );
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document.write( "the vertex of your equation is found by the equation:\r<BR>\n" );
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document.write( "x = -b/2a.\r<BR>\n" );
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document.write( "once your equation is in standard form, you can find the value of a, b, and c.\r<BR>\n" );
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document.write( "the standard form of a quadratic equation is ax^2 + bx + c = 0\r<BR>\n" );
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document.write( "your equation, in standard form, is -x^2 + 4x - 3 = 0.\r<BR>\n" );
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document.write( "that makes a = -1, b = 4, c = -3\r<BR>\n" );
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document.write( "a is the coefficient of the x^2 term.<BR>\n" );
document.write( "b is the coefficient of the x term.<BR>\n" );
document.write( "c is the coefficient of the constant term.\r<BR>\n" );
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document.write( "the formula for the vertex of a quadratic equation is x = -b/2a.\r<BR>\n" );
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document.write( "in your equation, that translates to x = - (-4) / 2 which becomes 4/2 which becomes 2.\r<BR>\n" );
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document.write( "the y value of the vertex is the value of y when x = 2.\r<BR>\n" );
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document.write( "that would be -x^2 + 4x - 3 = -(2^2) + 8 - 3 = -4 + 8 - 3 = 4 - 3 = 1.\r<BR>\n" );
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document.write( "that puts the vertex of your quadratic equation at (x,1) as confirmed by the graph.\r<BR>\n" );
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document.write( "the axis of symmetry of your graph is the x value of your vertex.\r<BR>\n" );
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document.write( "that makes the axis of symmetry of your graph at x = 2.\r<BR>\n" );
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document.write( "to prove that it is the axis of symmetry, you take a value of y and solve for x.\r<BR>\n" );
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document.write( "you can choose value of y = 0 to make it easy.\r<BR>\n" );
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document.write( "when y = 0, x = 1 and x = 3.\r<BR>\n" );
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document.write( "that makes the value of x equidistant from the axis of symmetry as it should be.\r<BR>\n" );
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document.write( "the y-intercept is the value of y when x = 0.\r<BR>\n" );
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document.write( "to find that, simply replace x with 0 in your equation to get y = -x^2 + 4x - 3 becomes y = 0 + 0 - 3 which becomes y = -3 as confirmed by the graph.\r<BR>\n" );
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document.write( "to plot the graph, you simply take values of x and solve for y and then plot the value of the (x,y) pairs on the graph.\r<BR>\n" );
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document.write( "a simple table below shows you what you will find.\r<BR>\n" );
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document.write( "<PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( " X	Y = -X^2 + 4X - 3\r\n" );
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document.write( "-2	-15\r\n" );
document.write( "-1	-8\r\n" );
document.write( "0	-3\r\n" );
document.write( "1	0\r\n" );
document.write( "2	1\r\n" );
document.write( "3	0\r\n" );
document.write( "4	-3\r\n" );
document.write( "5	-8\r\n" );
document.write( "6	-15\r\n" );
document.write( "</PRE>\r<BR>\n" );
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document.write( "Since the axis of symmetry is at x = 2, you can see that:\r<BR>\n" );
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document.write( "when x = 0 and x = 4, the value of y is equal to -3.\r<BR>\n" );
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document.write( "that's because 0 is 2 units away from 2 and 4 is also 2 units away from 2.\r<BR>\n" );
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document.write( "since the graph is symmetric about the value of x = 2, the value of y at those points equi-distant from the axis of symmetry will be the same.\r<BR>\n" );
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document.write( "same goes for y = -15 when x = 6 and x = -2.\r<BR>\n" );
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document.write( "same goes for y = -8 when x = 5 and x = -1.\r<BR>\n" );
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document.write( "back to factoring.\r<BR>\n" );
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document.write( "if you did not multiply both sides of the equation by -1, you can still factor it, only it's a little harder to see the factors.\r<BR>\n" );
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document.write( "i'll do it here just to show you.\r<BR>\n" );
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document.write( "your original equation is -x^2 + 4x - 3.\r<BR>\n" );
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document.write( "since the coefficient of the x^2 term is negative, one of your x values has to be negative, so you will get factors that look like:\r<BR>\n" );
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document.write( "(-x + a) * (x + b) = -x^2 + 4x - 3\r<BR>\n" );
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document.write( "a and b are the constant terms that you want to find.\r<BR>\n" );
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document.write( "since your constant term has to be -3, then you will multiply by 3 * -1, or you will multiply by -3 * 1\r<BR>\n" );
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document.write( "let's try -3 * 1.\r<BR>\n" );
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document.write( "you will get (-x - 3) * (x + 1)\r<BR>\n" );
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document.write( "that results in -x^2 - x -3x - 3 which results in -x^2 -4x - 3 which is not what we want.\r<BR>\n" );
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document.write( "let's try (-x + 3) * (x - 1)\r<BR>\n" );
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document.write( "that results in -x^2 - x + 3x - 3 which results in -x^2 + 2x - 3 which is not what we want.\r<BR>\n" );
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document.write( "let's try (-x + 1) * (x - 3)\r<BR>\n" );
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document.write( "that results in -x^2 + 3x + x - 3 which results in -x^2 + 4x - 3 which is what we want so we can stop here.\r<BR>\n" );
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document.write( "our factors become:\r<BR>\n" );
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document.write( "(-x+1) * (x-3) = 0\r<BR>\n" );
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document.write( "from the factor of (x-3) = 0, we derive x = 3.\r<BR>\n" );
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document.write( "frm the factor of (-x+1) = 0, we derive -x = -1 which results in x = 3 when we multiply both sides of the equation by -1.\r<BR>\n" );
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document.write( "whether we multiplied both sides of the quadratic equation in standard form by -1 in the beginning or not, we still wind up with the same factors.\r<BR>\n" );
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document.write( "it's just easier to see what the factors are by doing that.\r<BR>\n" );
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document.write( "you are left to do that or not at your discretion.\r<BR>\n" );
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document.write( "i always found it easier to solve the quadratic equation if the coefficient of the x^2 term was positive.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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