SOLUTION: Can you explain exactly how a Quadratic equation works and how you put it on a graph, i.e parabola?

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Question 476484: Can you explain exactly how a Quadratic equation works and how you put it on a graph, i.e parabola?
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the quadratic equation is of the form:
y = ax^2 + bx + c
a is the coefficient of the x^2 term
b is the coefficient of the x term
c is the constant
if a is positive, then the graph points down and opens up.
if a is negative, then the graph points up and opens down.
here's a graph of x^2 + 2x + 1
this one will point down and open up because the coefficient of the x^2 term is positive.
graph%28400%2C400%2C-5%2C5%2C-5%2C5%2Cx%5E2+%2B+2x+%2B+1%29
here's a graph of -x^2 + 2x + 1
this one will point up and open down because the coefficient of the x^2 term is negative.
graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C-x%5E2+%2B+2x+%2B+1%29
you can find the roots of the quadratic equation by use of the quadratic formula.
that formula is:
x = %28%28-b%29%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29
you can find the value of x for the min/max point of the quadratic equation by using the formula:
x = %28-b%29%2F%282a%29
you can find the value of y for the min/max point of the quadratic equation by using the formula:
y = f%28%28-b%29%2F%282a%29%29
f%28%28-b%29%2F%282a%29%29 means that, after you found the value of x for the min/max point, you replace x in the equation with that value for x and solve for y.
an example will show you how these formulas work.
we'll work with the equation:
y = x^2 - 3x - 10
to find the roots of that equation, we set the equation equal to 0 and use the quadratic equation to find the roots.
our equation becomes:
x^2 - 3x - 10 = 0
since this is now in standard form of ax^2 + bx + c = 0, we get:
a = 1
b = -3
c = -10
we plug those values into the quadratic formula to get:
x = %28%28-b%29%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29 becomes:
x = %28%28-%28-3%29%29%2B-sqrt%28%28-3%29%5E2-4%2A%281%2A%28-10%29%29%29%29%2F%282%2A1%29 which becomes:
x = %283%2B-sqrt%2849%29%29%2F%282%29
this leads to 2 possible values of x.
they are:
x = 5
x = 2
those are the values of x which lead to 0 for a value of y.
the roots of the quadratic equation are when the quadratic equation crosses the x-axis.
this happens when the value of y = 0.
we can find the min/max point of this equation by using the -b%2F2a formula to get the x value.
using that formula, we get:
x = -%28-3%29%2F%282%2A1%29 which makes x = (3/2).
the y value if found by getting f%28%28-b%2F2a%29%29 which becomesf%283%2F2%29 which means you replace x in the equation with (3/2) and solve for y.
the equation is:
y = x^2 -3x - 10
replacing x with (3/2) gets:
y = (3/2)^2 -3*(3/2) - 10 which gets:
y = 9/4 - 9/2 - 10 which gets:
y = 9/4 - 18/4 - 40/4 which gets:
-49/4 which is equivalent to
y = -12.25
to summarize what we have:
the roots of the equation are:
x = 5
x = 2
the min/max point of the equation is:
y = -49/4 when x = 3/2
the graph of this equation is shown below:

the horizontal line is at y = -49/4.
you can see that the min/max point intersects with that line which is the y value of the min/max point.
the vertical line is at x = 3/2.
you can see that the min/max point intersects with that line which is the x value of the min/max point.
the axis of symmetry of the quadratic equation is that vertical line.
the axis of symmetry of a quadratic equation means that the part of the graph to the left of that line is an exact mirror image of the part of the graph to the right of that line.