Calculating Half of a Parabolic Curve

A parabola can be thought of as a one-sided ellipse. Where a typical ellipse is closed and has two points within the shape called foci, a parabola is elliptical in shape but one focus is in infinity. An important feature of parabolas is that they are even functions, meaning that they are symmetric about their axis. The axis of symmetry of a parabola is called its vertex. Calculating half of a parabolic curve involves calculating the whole parabola and then taking points on only one side of the vertex.

Step 1

Ensure that the equation for the parabola is in the standard quadratic form f(x) = ax² + bx + c, where “a,” “b” and “c” are constant numbers and “a” is not equal to zero.

Step 2

Determine the direction that the parabola opens by examining the sign of “a.” If “a” is positive, then the parabola opens upward; if it is negative, the parabola opens downward.

Step 3

Find the x-coordinate of the vertex point for the parabola by substituting the “a” and “b” values into the expression: -b / 2a.

Step 4

Find the y-coordinate of the vertex point for the parabola by substituting the previously determined x-coordinate into the original quadratic equation and then solving the equation for y. For example, if f(x) = 3x² + 2x + 5 and the x-coordinate is known to be 4, then the initial equation becomes: f(x) = 3(4)² + 2(4) + 5 = 48 + 8 + 5 = 61. So the vertex point for this equation is (4,61).

Step 5

Find any x-intercepts of the equation by setting it to 0 and solving for x. If this method is not possible, substitute the “a,” “b” and “c” values into the quadratic equation ((-b ± sqrt(b² – 4ac)) / 2a).

Step 6

Find any y-intercepts by setting the x-value to 0 and solving for f(x). The resulting value is the y-intercept.

Step 7

Plot one half of the parabola by choosing x-values that are either less than the x-coordinate or greater than the x-coordinate of the vertex, but not both.

Step 8

Substitute these x-values into the original quadratic equations to determine the y-coordinate for each x-value.

Step 9

Plot the appropriate points, intercepts and vertex point on a Cartesian coordinate plane. Then connect the points with a smooth curve to complete the parabola half.