How to Calculate a Parabolic Curve

By Sky Smith
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Parabolas normally illustrate how a thrown object moves in relation to gravity, but they can apply to other areas of science as well. By the middle of Algebra I, most students should have first encountered quadratic equations and the parabolic curves they make. These U-shaped graphs can be calculated by following a few basic, though somewhat challenging, steps.

Organize your equation or function into standard form. This means to set up the equation so that it reads y = or f(x) = Ax^2 + Bx + C, where A, B and C are numbers and A does not equal zero.

Determine if your vertex is a minimum or maximum. If A is greater than zero, then you will have a minimum. If A is less than zero, you will have a maximum. Your vertex establishes the highest or lowest point on your graph and all other points will be symmetric about this vertex.

Calculate the x-value of your vertex. To do this, simplify the following formula: the opposite of B divided by (2 times A), or --B/(2A).

Calculate the y-value of your vertex. Taking the result of the x-value from step 3, substitute this value back into the original equation in place of "x". For instance, if f(x) = 2x^2 + x + 1 and your x-value of the vertex is 3, then f(3) = 2(3)^2 + (3) + 1 = 18 + 3 + 1 = 22, suggesting your y-value of the vertex is 22.

Plot your vertex using the x-value and y-value calculated from steps 3 and 4.

Calculate any x-intercepts. X-intercepts occur when the function, or equation, equals zero, so set your function equal to zero and solve. Alternatively, you can input your values of A, B and C into the following quadratic formula: [-- B +/- sqrt(B^2 -- 4AC) ] / (2A).

Calculate your y-intercept. That means to plug x = 0 into your function and solve for the resulting value of y.

Graph your x- and y-intercepts. Not all parabolas will have x-intercepts, but all parabolas will have one y-intercept and a vertex.

Calculate and plot additional points as necessary. You may need to plot one or two additional points to complete your graph. To do so, pick values of x like x = 1 or x = -2. Then substitute them into the function to find corresponding y-values. Keep in mind that your graph should look similar to a "U" symmetric about the vertex.