Graphing Polynomial Functions

By the end of this lesson, you will be able to sketch polynomial graphs using zeros (with multiplicity), $y$-intercept, end behavior, and turning points, and identify local and global features from graphs.

The Big Idea

Graphing a polynomial without a calculator is a synthesis of everything in this unit: end behavior (degree + leading coefficient), $x$-intercepts (zeros + multiplicity), $y$-intercept, and a rough count of turning points. These features, plotted in order, produce an accurate qualitative sketch.

The Complete Graphing Procedure

1. End behavior from degree and leading coefficient
2. $x$-intercepts from the zeros (and multiplicity for cross vs. bounce)
3. $y$-intercept by evaluating $f(0)$
4. Turning points — at most $n-1$ for degree $n$
5. Sign analysis — determine whether $f$ is positive or negative in each interval between zeros
6. Sketch — connect the information

Example 1 — Step-by-Step Sketch

Graph $f(x) = (x+2)(x-1)^2(x-4)$

Step 1 — End behavior:

Degree: $1+2+1 = 4$ (even); leading coefficient: positive (all factors have positive leading terms).

Both tails up.

Step 2 — Zeros:

$x = -2$ (mult. 1) → crosses

$x = 1$ (mult. 2) → bounces

$x = 4$ (mult. 1) → crosses

Step 3 — $y$-intercept:

$f(0) = (2)(1)(-4) = -8$ → $(0, -8)$

Step 4 — Sign analysis (between zeros):

• $x < -2$: all tails up from the right → positive (both tails up, degree 4 positive)
• $-2 < x < 1$: test $x = 0$: $f(0) = -8 < 0$ → negative
• $1 < x < 4$: test $x = 2$: $f(2) = (4)(1)(-2) = -8 < 0$ → negative (bounce at $x=1$ keeps same sign)
• $x > 4$: positive (right tail up)

Sketch: comes from upper-left, crosses $x$-axis at $-2$, dips below, bounces off at $x = 1$, continues below, crosses at $x = 4$, rises to upper-right.

Example 2 — Identifying Graph Features

$g(x) = -2x^3 + 6x^2 - 4x$.

Factor: $g(x) = -2x(x^2 - 3x + 2) = -2x(x-1)(x-2)$

Zeros: $x = 0, 1, 2$ (all mult. 1 → all cross)

Degree 3, negative leading → left up, right down.

$y$-intercept: $g(0) = 0$ (on the $x$-axis)

Sign test ($x = 0.5$): $g(0.5) = -2(0.5)(−0.5)(−1.5) = -2(0.5)(0.75) = -0.75 < 0$