document.write( "Question 168162: If you are looking at a graph of a quadratic equation, how do you determine where the solutions are? \n" ); document.write( "
Algebra.Com's Answer #123956 by gonzo(654)\"\" \"About 
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any point on the graph is a solution of the equation of that graph.
\n" ); document.write( "example:
\n" ); document.write( "if your equation is y = x^2 + + x + 1, then when you look at the graph and see a point on the graph, you plot what the x value of that point is and the y value of that point is and you have a solution of the graph.
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\n" ); document.write( "if it is a quadratic equation, where the graph crosses the x axis tells you what the roots are of the equation.
\n" ); document.write( "these are called the x intercepts.
\n" ); document.write( "if the graph doesn't cross the x-axis then it doesn't have any real roots.
\n" ); document.write( "since a quadratic is in the form of ax^2 + bx + c,
\n" ); document.write( "then a, b, and c tell you a few things about the graph also.
\n" ); document.write( "-b/2a is the x value of the minimum / maximum point on the graph.
\n" ); document.write( "if the graph is pointing upwards, then -b/2a is the x value of a maximum point.
\n" ); document.write( "if the graph is pointing downward, then -b/2a is the x value of a minimum point.
\n" ); document.write( "the tails of the graph go in the opposite direction of where the head is pointing.
\n" ); document.write( "if x^2 is positive, then the graph is pointing downward (head down, tails up).
\n" ); document.write( "if x^2 is negative which can only be if is it multiplied by a negative number, such as -x^2, then the graph is pointing upward (head up, tails down).
\n" ); document.write( "the y intercept is found when x = 0 which is the intersection of the x axis with the y axis.
\n" ); document.write( "here's a graph of x^2 - 7x + 10
\n" ); document.write( "the roots of the equation are x = 2, and x = 5
\n" ); document.write( "since x^2 is positive, the graph points down (head down, tails up).
\n" ); document.write( "the x value of the minimum point is -b/2a = -(-7)/2 = 7/2 = 3.5
\n" ); document.write( "the y value of the minimum point is found by substituting x = 3.5 into the equation.
\n" ); document.write( "that value becomes (3.5)^2 - 7(3.5) + 10 = -2.25.
\n" ); document.write( "the minimum point on the graph is (3.5,-2.25) which can be seen on the graph.
\n" ); document.write( "if the graph is symmetric about the y value, then the axis of symmetry would be the x coordinate of the vertex which is at the minimum or maximum point on the graph.
\n" ); document.write( "in this graph the axis of symmetry would be x = 3.5.
\n" ); document.write( "the same y value would give 2 values for x which are equidistant from the axis of symmetry.
\n" ); document.write( "one example of symmetry is already solved where y = 0.
\n" ); document.write( "x = 2, and x = 5.
\n" ); document.write( "if the axis of symmetry is 3.5, then:
\n" ); document.write( "3.5 - 2 = 1.5
\n" ); document.write( "5 - 3.5 = 1.5
\n" ); document.write( "both these points are equidistant from the axis of symmetry when x = 0.
\n" ); document.write( "take any other 2 x value on the graph that are equidistant from 3.5 and the y values should be the same.
\n" ); document.write( "try 3.5 + 5 = 8.5, and 3.5 - 5 = -1.5
\n" ); document.write( "when x = 8.5, y = 22.75
\n" ); document.write( "when x = -1.5, y = 22.75
\n" ); document.write( "the graph is symmetric about the value of x = 3.5.
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