Functions
AI Assistant

Function Arithmetic II

Quadratic operations, division, and the mystery of the hole

In Function Arithmetic I, you learned that multiplying two lines gives a parabola (degree 1 × 1 = 2). Now go the other direction: what happens when you divide a quadratic by a line?

The degree drops back down — but something unexpected appears: a hole in the graph where division by zero occurs. This is your first encounter with discontinuity, one of the most important ideas in mathematics.

Ask the AI anything — try "Why is there a hole at x = 2?" or "What value does it approach?"

Graph

Popular Lessons

Word Problem Solver Visual Word Problem Solver Function Root Finder

FAQ

What happens when you add two quadratic functions?
Adding two quadratics gives another quadratic — unless the x² terms cancel out. For example, x² + (-x² + 4) = 4, a constant. When leading terms cancel, the degree drops.
What happens when you divide a quadratic by a linear function?
If the linear function is a factor of the quadratic, you get a simpler function (degree 2 ÷ 1 = degree 1). But there is a hole at the x-value where the divisor equals zero, because division by zero is undefined.
What is a discontinuity?
A discontinuity is a point where a function is not defined or has a break. A removable discontinuity (hole) occurs when a factor cancels in a fraction — the function approaches a value but never reaches it at that specific point.
What is the degree of a function?
The degree is the highest power of x. Linear functions have degree 1, quadratics have degree 2. Multiplying functions adds degrees, dividing subtracts them. This pattern governs all of polynomial algebra.

Related Topics

Function Arithmetic IQuadratic EquationsFactoringRational FunctionsLimitsDomain & Range