How this instrument works
A power function has the form y = a·xᵇ, where a and b are fixed constants and x is the variable being fed in. It's easy to confuse with an exponential function, y = a·bˣ, but the two put the variable in completely different positions — a power function's variable is the BASE being raised to a fixed exponent, while an exponential function's variable sits in the EXPONENT itself, applied to a fixed base. The two behave very differently as x grows large, exponential growth eventually outpacing any fixed-exponent power function no matter how large that exponent is.
The exponent b shapes the function's whole personality: b = 1 gives a plain straight line, b = 2 gives a parabola, b = 3 gives a cubic curve, and a NEGATIVE b turns the function into a curve that shrinks toward zero as x grows, since a negative exponent means dividing by increasing powers of x rather than multiplying by them.
Power functions with a whole-number exponent are the building blocks of every polynomial — a polynomial is nothing more than several power functions with different exponents added together, each one's own coefficient acting as this formula's a. Understanding a single power function's shape is the first step toward understanding how a full polynomial's shape emerges from combining several of them.
- Enter the coefficient into the Coefficient (a) field.
- Enter the fixed exponent into the Exponent (b) field.
- Enter the value to evaluate the function at into the x field.
- Read y: the sheet computes a·xᵇ directly.
Worked example — y = 2x³ at x = 4
The power function y = 2x³, evaluated at x = 4, gives y = 2 × 4³ = 2 × 64 = 128 — the exponent 3 is fixed as part of the function's definition, while x is the variable input being plugged in, distinct from an exponential function where 4 might instead sit in the exponent position.
With exponent b = 0, the function y = 1·x⁰ collapses to y = 1 regardless of x's value, since any nonzero number raised to the zeroth power is 1. And y = 3x², evaluated at x = −2, gives y = 3 × 4 = 12 — a negative input squared still lands on a positive result, since squaring always erases the sign.
Questions
What is a power function?
A function of the form y = a·xᵇ, where a and b are fixed constants and x is the variable input, raised to that fixed exponent b. The variable sits in the BASE position, not the exponent.
How is a power function different from an exponential function?
A power function, y = a·xᵇ, has the variable in the base with a fixed exponent. An exponential function, y = a·bˣ, has the variable in the EXPONENT with a fixed base. The two grow at fundamentally different rates as x increases — exponential growth eventually overtakes any power function.
How does the exponent shape the function's graph?
b = 1 gives a straight line, b = 2 gives a parabola, b = 3 gives a cubic curve, and larger whole-number exponents give progressively steeper, flatter-near-zero curves. A negative exponent instead makes the function shrink toward zero as x grows larger.
Are polynomials made of power functions?
Yes — a polynomial is a sum of several power functions with different whole-number exponents, each with its own coefficient. A quadratic ax² + bx + c, for instance, is really three power functions (exponents 2, 1, and 0) added together.
What happens with a negative base and a fractional exponent?
This can produce an undefined (non-real) result, since raising a negative number to a fractional power often requires a complex number — this calculator handles real-valued inputs where the result stays within the real numbers.