Power function

Definition

A power function is a function of the form $f(x) = ax^n$ where $a$ and $n$ are constants, $a \neq 0$, and $n$ is a real number. Power functions are a basic type of polynomial function when $n$ is a non-negative integer.

5 Must Know Facts For Your Next Test

1. The graph of a power function with an even exponent ($n$) opens upwards or downwards, resembling a parabola.
2. The graph of a power function with an odd exponent ($n$) has opposite behavior on each side of the y-axis, resembling an S-curve.
3. If the exponent ($n$) is positive, the function grows as x moves away from zero; if negative, it approaches zero as x moves away from zero.
4. The coefficient ($a$) affects the vertical stretch or compression and direction (upward for positive, downward for negative).
5. Power functions are continuous and smooth curves without breaks or corners.

Review Questions

• What is the general form of a power function?
• How does the graph of a power function change when the exponent is even versus odd?
• Explain how the coefficient in front of the variable affects the shape and direction of a power function.

Related terms

Polynomial Function:

A polynomial function is a sum of terms consisting of variables raised to non-negative integer powers multiplied by coefficients.

Rational Function:

A rational function is defined as the ratio of two polynomials, where both numerator and denominator are polynomials.

Exponent:

An exponent indicates how many times to multiply a base number by itself in expressions like $x^n$.