Graphing Polynomial Functions: Steps

Graphing polynomial functions by hand uses a few key features instead of plotting dozens of points. Find the zeros and their multiplicities, the y-intercept and the end behavior. Then check the sign between zeros and connect everything with one smooth curve that has at most $n - 1$ turns.

Before this lesson: You should know End Behavior of Polynomials, Fundamental Theorem of Algebra, Solving Polynomial Equations and X-Intercepts and Y-Intercepts.

What Shapes a Polynomial Graph

Polynomial graphs are smooth and unbroken, with no corners or gaps. Four features tell you almost everything about the shape.

Feature	How to find it	What it shows
Zeros	Set each factor equal to 0	Where the graph meets the x-axis
Multiplicity	Count how many times each factor appears	Odd: crosses the axis. Even: touches and turns back.
y-intercept	Find $f(0)$	Where the graph meets the y-axis
End behavior	Degree and sign of the leading coefficient	Where the two ends point

A polynomial of degree $n$ has at most $n - 1$ turning points, where the graph changes from rising to falling or the other way.

Graph of f(x) = (x + 2)(x − 1)^2(x − 3). Both ends point up. The curve crosses the x-axis at −2, dips to a low point, rises to touch the x-axis at 1, dips again and crosses at 3. The y-intercept (0, −6) is marked. Labels show crosses at −2 and 3, touches at 1. — Degree 4 with a positive leading coefficient: both ends rise. It crosses at −2 and 3 and touches at 1.

Steps for Graphing a Polynomial

Factor the polynomial completely.
Find the zeros and their multiplicities.
Find the y-intercept, $f(0)$ .
Find the end behavior from the leading term.
Test one x-value between each pair of zeros to see if the graph is above or below the axis.
Plot the points and draw one smooth curve that matches the ends, crosses or touches at each zero, and turns at most $n - 1$ times.

Worked Examples

Example 1: A cubic

Graph $f(x) = x^3 - x^2 - 6x$ .

Factor out $x$ (see factoring out the GCF): $f(x) = x(x^2 - x - 6)$
Factor the trinomial (see factoring trinomials): $f(x) = x(x - 3)(x + 2)$
Zeros: −2, 0 and 3, each with multiplicity 1, so the graph crosses at each one.
y-intercept: $f(0) = 0$ , the point $(0, 0)$
End behavior (see end behavior of polynomials): leading term $x^3$ means down on the left, up on the right.
Test $x = -1$ : $f(-1) = (-1)(-4)(1) = 4$ , above the axis
Test $x = 1$ : $f(1) = (1)(-2)(3) = -6$ , below the axis
Sketch: rise from the lower left, cross at −2, peak near $x = -1$ , cross at 0, dip near $x = 2$ , cross at 3, then rise.

Answer: An S-shaped curve through $(-2, 0)$ , $(0, 0)$ and $(3, 0)$ , with 2 turning points (degree 3 allows at most 2).

Example 2: A quartic with a touching zero

Graph $f(x) = (x + 2)(x - 1)^2(x - 3)$ . This is the graph above.

Zeros: −2 (multiplicity 1), 1 (multiplicity 2) and 3 (multiplicity 1) (see fundamental theorem of algebra).
The graph crosses at −2 and 3 and touches at 1.
y-intercept: $f(0) = (2)(-1)^2(-3)$
Evaluate: $(2)(1)(-3) = -6$ , the point $(0, -6)$
Degree: $1 + 2 + 1 = 4$ . The leading coefficient is $1 \cdot 1 \cdot 1 = 1$ , positive. So both ends point up.
Test $x = -1$ : $f(-1) = (1)(-2)^2(-4) = (1)(4)(-4) = -16$ , below the axis
Test $x = 2$ : $f(2) = (4)(1)^2(-1) = -4$ , below the axis
Sketch: come down from the upper left, cross at −2, dip near −16, rise to touch at 1, dip near −4, cross at 3, rise to the upper right.

Answer: A W-shaped curve with 3 turning points (degree 4 allows at most 3).

Example 3: A negative leading coefficient

Graph $f(x) = -x^3 + 4x$ .

Factor out $-x$ : $f(x) = -x(x^2 - 4)$
Factor the difference of squares (see factoring the difference of squares): $f(x) = -x(x - 2)(x + 2)$
Zeros: −2, 0 and 2, each crossing.
y-intercept: $f(0) = 0$
End behavior: leading term $-x^3$ means up on the left, down on the right.
Test $x = -1$ : $f(-1) = -(-1)^3 + 4(-1) = 1 - 4 = -3$ , below
Test $x = 1$ : $f(1) = -(1)^3 + 4(1) = -1 + 4 = 3$ , above

Answer: The curve falls from the upper left, crosses at −2, dips to about −3, rises through the origin to about 3, and falls through 2 to the lower right.

Example 4: Write the equation from a graph

A degree-3 graph crosses the x-axis at −3, touches it at 2, and has y-intercept −24. Write its equation.

Crossing at −3 gives the factor $x + 3$ . Touching at 2 gives $(x - 2)^2$ .
Write the form with an unknown leading coefficient $a$ : $f(x) = a(x + 3)(x - 2)^2$
Use the y-intercept: $f(0) = a(3)(-2)^2 = -24$
Simplify: $12a = -24$
Divide both sides by 12: $a = -2$

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

Example 5: Word problem with an open box

An open box is made from a 20-inch by 12-inch sheet by cutting $x$ -inch squares from each corner. Its volume is $V(x) = x(20 - 2x)(12 - 2x)$ . Sketch the graph, decide which part makes sense, and find the best whole-inch cut.

Zeros: $x = 0$ , $20 - 2x = 0$ gives $x = 10$ , and $12 - 2x = 0$ gives $x = 6$ .
Leading term: $x(-2x)(-2x) = 4x^3$ , so down on the left and up on the right.
A cut must be positive and leave a bottom, so only $0 \lt x \lt 6$ makes sense.
$V(1) = 1 \cdot 18 \cdot 10 = 180$
$V(2) = 2 \cdot 16 \cdot 8 = 256$
$V(3) = 3 \cdot 14 \cdot 6 = 252$
$V(4) = 4 \cdot 12 \cdot 4 = 192$
$V(5) = 5 \cdot 10 \cdot 2 = 100$

Answer: On $0 \lt x \lt 6$ the graph rises to a peak and falls back to 0. The best whole-inch cut is 2 inches, which gives 256 cubic inches. The true peak is between 2 and 3 inches.

Common Mistakes

Mistake	Example of the error	Fix
Crossing at an even zero	Drawing $(x - 1)^2$ crossing the axis at 1	Even multiplicity touches the axis and turns back.
Wrong end behavior	Using $+5x$ for $5x - 3x^4$	Use the highest-degree term, $-3x^4$ : both ends down.
Wrong y-intercept from factored form	$f(x) = -2(x + 3)(x - 2)^2$ has y-intercept 12	Include the −2: $f(0) = -2(3)(4) = -24$ .
Too many turns	A cubic drawn with 3 turns	A degree- $n$ graph has at most $n - 1$ turning points.
Guessing the sign between zeros	Drawing the curve above the axis between −2 and 0 without checking	Test a point, like $f(-1)$ , to see if it is above or below.

Practice Problems

For $f(x) = (x - 4)(x + 1)^2$ , give the zeros, how the graph behaves at each, the y-intercept and the end behavior.
For $f(x) = x^3 - 4x^2$ , give the zeros with multiplicities and the end behavior.
Find the y-intercept and end behavior of $f(x) = -(x - 1)(x + 3)(x - 2)$ .
What is the greatest number of turning points of $f(x) = 5x^4 - x + 2$ ?
Find the zeros and y-intercept of $f(x) = x^4 - 5x^2 + 4$ .
For $f(x) = (x + 1)(x - 2)(x - 5)$ , is the graph above or below the x-axis between −1 and 2? Between 2 and 5?
A cubic has zeros −1 (multiplicity 1) and 3 (multiplicity 2) and passes through $(0, 18)$ . Write its equation.
A cubic touches the x-axis at −2, crosses at 1, and falls to the right. Write a possible equation.
For $f(x) = x(x - 2)(x + 3)$ , find $f(-1)$ and $f(1)$ .
An open box is made from a 16-inch by 10-inch sheet by cutting $x$ -inch corner squares: $V(x) = x(16 - 2x)(10 - 2x)$ . Find the zeros, the sensible domain and the best whole-inch cut.

Show answers

Zeros: 4 (multiplicity 1, crosses) and −1 (multiplicity 2, touches)
y-intercept: $f(0) = (-4)(1)^2 = -4$
Leading term $x^3$ : down on the left, up on the right
Factor out $x^2$ : $f(x) = x^2(x - 4)$
Zeros: 0 (multiplicity 2, touches) and 4 (multiplicity 1, crosses)
Leading term $x^3$ : down on the left, up on the right
y-intercept: $f(0) = -(-1)(3)(-2)$
Multiply: $-(6) = -6$
Leading term $-x^3$ : up on the left, down on the right
The degree is 4
Answer: at most $4 - 1 = 3$ turning points
Factor: $(x^2 - 1)(x^2 - 4)$
Factor again: $(x - 1)(x + 1)(x - 2)(x + 2)$
Zeros: $\pm 1, \pm 2$ , each crossing
y-intercept: $f(0) = 4$
Test $x = 0$ : $(1)(-2)(-5) = 10$ , so above the axis between −1 and 2
Test $x = 3$ : $(4)(1)(-2) = -8$ , so below the axis between 2 and 5
Form: $f(x) = a(x + 1)(x - 3)^2$
Use $(0, 18)$ : $a(1)(-3)^2 = 18$
Simplify: $9a = 18$
Divide by 9: $a = 2$
Answer: $f(x) = 2(x + 1)(x - 3)^2$
Touching at −2 needs $(x + 2)^2$ ; crossing at 1 needs $(x - 1)$
Falling to the right with odd degree needs a negative leading coefficient
Answer: $f(x) = -(x + 2)^2(x - 1)$ (any negative $a$ works)
$f(-1) = (-1)(-3)(2) = 6$
$f(1) = (1)(-1)(4) = -4$
Zeros: $x = 0$ , $x = 8$ and $x = 5$
Sensible domain: $0 \lt x \lt 5$
$V(1) = 1 \cdot 14 \cdot 8 = 112$
$V(2) = 2 \cdot 12 \cdot 6 = 144$
$V(3) = 3 \cdot 10 \cdot 4 = 120$
$V(4) = 4 \cdot 8 \cdot 2 = 64$
Answer: the best whole-inch cut is 2 inches, for 144 cubic inches

FAQ

How do you graph a polynomial function?

Factor it to find the zeros and multiplicities, find the y-intercept, and use the leading term for end behavior. Test points between zeros, then draw a smooth curve through everything.

How does multiplicity change the graph?

At a zero with odd multiplicity, the graph crosses the x-axis. At a zero with even multiplicity, it touches the axis and turns around. Higher multiplicities also flatten the graph near the zero.

How many turning points can a polynomial have?

A polynomial of degree $n$ has at most $n - 1$ turning points. A cubic can have 0 or 2, and a quartic can have 1 or 3.

Can you find the exact high and low points by hand?

Not with Algebra 2 tools, except in special cases. Use a table of values or a graphing calculator to estimate them. Calculus gives exact answers.

Summary

Factor to find the zeros and their multiplicities.
Odd multiplicity crosses the x-axis; even multiplicity touches it.
Find the y-intercept with $f(0)$ and the end behavior from the leading term.
Test a point between zeros to see whether the graph is above or below.
A degree- $n$ graph has at most $n - 1$ turning points.

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