Advanced Polynomial Graph Analysis

By the end of this lesson, you will be able to perform a complete graphical analysis of any polynomial function, discuss local/global extrema, inflection points, and increasing/decreasing intervals at a pre-calculus level.

The Big Idea

A complete polynomial analysis connects algebra (zeros, multiplicity, end behavior) with function behavior (increasing/decreasing intervals, local extrema). At the pre-calculus level, we determine these using test values in each interval between zeros — a technique that foreshadows the first derivative test in calculus.

Complete Analysis Framework

1. Degree and leading coefficient → end behavior
2. Find all zeros (rational root theorem, factoring, quadratic formula)
3. Multiplicity at each zero → cross or bounce
4. $y$-intercept → evaluate at $x = 0$
5. Sign chart → determine where $f > 0$ and $f < 0$
6. Local extrema → at turning points between zeros (use sign chart)
7. Sketch combining all features

Example 1 — Complete Analysis

$f(x) = x^3 - 3x + 2$

Zeros: Test $x = 1$: $1 - 3 + 2 = 0$ ✓. Divide: $(x-1)(x^2 + x - 2) = (x-1)(x+2)(x-1) = (x-1)^2(x+2)$

Zeros: $x = 1$ (mult. 2, bounce), $x = -2$ (mult. 1, cross)

End behavior: degree 3, positive leading → left down, right up.

$y$-intercept: $f(0) = 2$

Sign chart:

• $x < -2$: $f < 0$ (from end behavior)
• $-2 < x < 1$: test $x = 0$: $f(0) = 2 > 0$
• $x > 1$: $f > 0$ (right tail up, bounces at 1)

Local minimum: At $x = 1$ (bounce) — $f(1) = 0$. Local max somewhere between $-2$ and $1$.

Turning Points

A degree-$n$ polynomial has at most $n-1$ turning points.

To find where turning points occur (without calculus): use the sign chart. A turning point is where $f$ changes from increasing to decreasing or vice versa.

Between $x = -2$ and $x = 1$: $f$ increases then decreases (since it rises from $f(-2) = 0$ to a local max, then falls back to $f(1) = 0$).