A quadratic function or equation has the form f(x) = ax^{2} + bx + c. It contains three terms:

- ax
^{2}= the**quadratic term**(a is the leading coefficient). This term is always raised to 2, so is sometimes called the*squared term*. - bx = the linear term,
- c = the constant term.

For example, for the function f(x) = 5x^{2} + 4x – 2, the quadratic term is 5x^{2}. In the expression f(x) = x^{2} + 2, it’s x^{2}. 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.

It’s called a *quadratic* term because it’s the term that makes the expression a quadratic function: take away the “ax^{2}” 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:

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.

**CITE THIS AS:**

**Stephanie Glen**. "Quadratic Term: Definition, Examples" From

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