SOLUTION: If you are looking at a graph of a quadratic equation, how do you determine where the solutions are?
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Question 168162: If you are looking at a graph of a quadratic equation, how do you determine where the solutions are?
Answer by gonzo(654) (Show Source): You can put this solution on YOUR website!
any point on the graph is a solution of the equation of that graph.
example:
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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if it is a quadratic equation, where the graph crosses the x axis tells you what the roots are of the equation.
these are called the x intercepts.
if the graph doesn't cross the x-axis then it doesn't have any real roots.
since a quadratic is in the form of ax^2 + bx + c,
then a, b, and c tell you a few things about the graph also.
-b/2a is the x value of the minimum / maximum point on the graph.
if the graph is pointing upwards, then -b/2a is the x value of a maximum point.
if the graph is pointing downward, then -b/2a is the x value of a minimum point.
the tails of the graph go in the opposite direction of where the head is pointing.
if x^2 is positive, then the graph is pointing downward (head down, tails up).
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).
the y intercept is found when x = 0 which is the intersection of the x axis with the y axis.
here's a graph of x^2 - 7x + 10
the roots of the equation are x = 2, and x = 5
since x^2 is positive, the graph points down (head down, tails up).
the x value of the minimum point is -b/2a = -(-7)/2 = 7/2 = 3.5
the y value of the minimum point is found by substituting x = 3.5 into the equation.
that value becomes (3.5)^2 - 7(3.5) + 10 = -2.25.
the minimum point on the graph is (3.5,-2.25) which can be seen on the graph.
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.
in this graph the axis of symmetry would be x = 3.5.
the same y value would give 2 values for x which are equidistant from the axis of symmetry.
one example of symmetry is already solved where y = 0.
x = 2, and x = 5.
if the axis of symmetry is 3.5, then:
3.5 - 2 = 1.5
5 - 3.5 = 1.5
both these points are equidistant from the axis of symmetry when x = 0.
take any other 2 x value on the graph that are equidistant from 3.5 and the y values should be the same.
try 3.5 + 5 = 8.5, and 3.5 - 5 = -1.5
when x = 8.5, y = 22.75
when x = -1.5, y = 22.75
the graph is symmetric about the value of x = 3.5.
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