Calculus How To

Linear Term: Definition, Examples

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A linear term has a degree of 1. For example, 5x, -2x and x are all linear terms. These terms are x to the first degree (X1), where the “1” isn’t written (because any number to the first power is just that number).

Terms that are not linear are called non-linear terms. The most common one you’ll come across in calculus is the quadratic term.

Quadratic Term, Linear Term and Constant Term

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A quadratic equation has the form f(x) = ax2 + bx + c, which contains three terms:


  • ax2 = the quadratic term,
  • bx = the linear term,
  • c = the constant term.

For example, for the function f(x) = 9x2 + 3x – 5, the linear term is 3x.

Not all quadratic functions have linear terms. For example, 10x2 – 5 = 0.

Coefficient of Linear Terms

The “3” in the above equation is the coefficient , and the “x” is the variable.

It’s possible to have more than one coefficient of a linear term. For example, the coefficient here:
f(x) = 9x2 + 3bx – 5
is 3b.

How It Affects the Shape of Parabolas

Changing the linear term affects the vertex position, both horizontally (along the x-axis) and vertically (along the y-axis).

Increasing “b” moves the parabola down and to the left:

linear term

The parabola moves down and to the left with increasing values of the linear term.



Decreasing “b” moves the parabola down and to the right:
parabola and linear term

The parabola shifts down and to the right with decreasing values for the linear term.



This may come as a surprise: decreasing “b” moves the graph towards the positive direction on the x-axis, and increasing it will move it in the negative x-direction.

References

Graphing calculator on Desmos.
Quadratic Functions. Retrieved November 5, 2019 from: https://math.dartmouth.edu/opencalc2/cole/lecture3.pdf

CITE THIS AS:
Stephanie Glen. "Linear Term: Definition, Examples" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/calculus-definitions/linear-term-definition-examples/
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