Calculus How To

Quadratic Term: Definition, Examples

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A quadratic function or equation has the form f(x) = ax2 + bx + c. It contains three terms:


For example, for the function f(x) = 5x2 + 4x – 2, the quadratic term is 5x2. In the expression f(x) = x2 + 2, it’s x2. The sign of the leading coefficient affects the shape of the parabola:


  • A positive coefficient will result in a “U” shape,
  • A negative coefficient will result in a “∩” shape.

sign of the leading coefficient

Positive coefficient (red) and negative (blue). Graphed with Desmos.com.

It’s called a quadratic term because it’s the term that makes the expression a quadratic function: take away the “ax2” and you’re left with a linear function bx + c.

Quadratic Term Example

Projectile motion can be described by, in part, a quadratic term. For example, the following graph shows the path of a baseball:
quadratic term

The expression needed to fully explain the baseball’s path has three parts:

  • The quadratic term represents the ball’s motion due to gravity. That’s what gives the path the distinctive quadratic (i.e. humped) shape: acceleration slows down as gravity pulls on the ball. Eventually the ball reaches its peak height, then gravity slows the ball down further until it eventually hits the ground.
  • The linear term represents the vertical (upwards) motion of the ball after it is hit by the bat. This part of the expression would be a straight line, except that gravity bends it downwards.
  • The constant term represents the ball’s initial height when hit. For a baseball, this would be roughly 2-3 ft, depending on the height of the player.

Next: Quadratic approximation.

References

Saveliev, P. (2019). Calculus Illustrated: Volume 1: Precalculus.
Desmos graphing calculator.

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Stephanie Glen. "Quadratic Term: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/quadratic-term-definition-examples/
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