How to Calculate the Max of a Parabola


One of the many ways scientists use parabolas is to model the motion of projectiles. If you shoot an object into the air at any angle, the height of the object as a function of time is a parabolic equation. The maximum value of the parabola is the point at which the object reaches its highest point. You can use calculus to determine the coordinates of the parabola's max. The first coordinate, t, is the time at which the projectile reaches its maximum height, and the second coordinate, y, is the maximum height.

  • Take the derivative of the parabolic equation using the rules for differentiation. Since a parabola is a second-degree polynomial, you use the power rule. For instance, suppose the parabola's equation is

    y = -4.9t^2 + 24.5t + 34.2 where y is the height measured in meters and t is the time measured in seconds. You calculate the derivative as y' = -9.8t + 24.5.

  • Set the derivative equation equal to zero and solve for t. This is the t-coordinate of the parabola's max. In this example, it is the time at which the object is at its highest point in the air. For instance, solving -9.8t + 24.5 = 0 gives you t = 2.5, which means the object is at its highest at 2.5 seconds.

  • Plug the t value into the original parabolic equation to compute the y-coordinate of the max. Using t = 2.5, you calculate y = -4.9(2.5^2) + 24.5(2.5) + 34.2, which equals 64.825. The maximum height of the projectile is 64.825 meters.

Tips & Warnings

  • You can apply these steps to find the max of any parabola equation, whether it models projectile motion or another mathematical process.

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