All Labs

Polynomial & Rational Functions Lab

Build polynomials up to degree 5, find their roots and turning points, and analyze end behavior. Switch to rational functions to discover vertical asymptotes, horizontal asymptotes, oblique asymptotes, and holes in the graph.

Guided Experiment: End Behavior & Degree

How does the degree of a polynomial (even vs odd) and the sign of its leading coefficient affect the end behavior of the graph?

Write your hypothesis in the Lab Report panel, then click Next.

Function Graph

CurveRoot (odd mult.)Turning point / Even mult.

Controls

Coefficients
Degree

Analysis Results

Function
f(x) = x² + x − 6
Degree
2
Leading Coefficient
1
End Behavior
x→−∞x \to -\infty +∞
x→+∞x \to +\infty +∞
Roots (x-intercepts)
x = 2.0000crosses x-axis
x = -3.0000crosses x-axis
Turning Points
(-0.5000, -6.2500)local min
y-intercept
(0, -6.0000)

Data Table

(0 rows)
#FunctionDegreeRootsEnd BehaviorTurning PointsAsymptotes
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0 / 500
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Reference Guide

Polynomial End Behavior

The end behavior of a polynomial is determined by its degree and leading coefficient.

f(x)=anxn+an−1xn−1+⋯+a0f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0

If n is even and the leading coefficient is positive, both ends go to +infinity. If n is odd and positive, the left end goes to negative infinity and the right end goes to positive infinity. A negative leading coefficient flips everything.

Finding Roots

A root (or zero) of a polynomial is a value of x where f(x) = 0. These are the x-intercepts.

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Root multiplicity matters. A root with odd multiplicity (1, 3, ...) means the graph crosses the x-axis. A root with even multiplicity (2, 4, ...) means the graph touches and bounces off.

Vertical & Horizontal Asymptotes

For rational functions f(x) = p(x)/q(x), vertical asymptotes occur where q(x) = 0 and the factor does not cancel.

If deg⁡(p)<deg⁡(q), then HA: y=0\text{If } \deg(p) < \deg(q), \text{ then HA: } y = 0

If degrees are equal, the horizontal asymptote is the ratio of leading coefficients. If the numerator degree exceeds the denominator by exactly 1, there is an oblique (slant) asymptote found via polynomial long division.

Holes in Rational Functions

A hole (removable discontinuity) occurs when a factor cancels between numerator and denominator.

(x−2)(x+1)(x−2)(x−3)→hole at x=2\frac{(x-2)(x+1)}{(x-2)(x-3)} \to \text{hole at } x = 2

At a hole, the function is undefined, but the limit exists. The y-coordinate of the hole is found by evaluating the simplified function at that x-value. Holes appear as open circles on the graph.