Parabola equations are written in the standard form of y=ax^2+bx+c. This form can tell you if the parabola opens up or down and, with a simple calculation, can tell you what the axis of symmetry is. While this is a common form to see an equation for a parabola in, there is another form that can give you a little more information about the parabola. The vertex form tells you the vertex of the parabola, which way it opens, and whether it is a wide or narrow parabola.
Using the standard equation of y=ax^2+bx+c, find the x value of the vertex point by plugging the a and b coefficients into the formula x= -b/2a.
For example:
y=3x^2+6x+8 x= -6/(2*3) = -6/6 = -1
Substitute the found value of x into the original equation to find the value of y.
y= 3(-1)^2+6(-1)+8 y= 3-6+8 y= 5
The values of x and y are the coordinates of the vertex. In this case, the vertex is at (-1,5).
Insert the vertex coordinates into the equation y= a(x-h)^2 + k, where h is the x-value and k is the y-value. The value of a comes from the original equation.
y = 3(x+1)^2+5 This is the vertex form of the parabola's equation.
(The h is a +1 in the equation because a negative in front of the -1 makes it positive.)
To convert the vertex form back to standard form, simply square the binomial, distribute a, and add the constants.
y=3(x+1)^2+5 y=3(x^2+2x+1)+5 y=3x^2+6x+3+5 y=3x^2+6x+8
This is the original standard form of the equation.
If a is positive, the parabola opens up. If a is negative, the parabola opens down. If |a|>1, the parabola is wide. If |a|<1, the parabola is narrow.
Watch the negative signs. Forgetting a negative is one of the most common mistakes. Copy the original problem carefully. Another common mistake is miscopying the original problem.
