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?"

Related Topics

Function Arithmetic IQuadratic EquationsFactoringRational FunctionsLimitsDomain & Range

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
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.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.