Graphing Polynomials

Main idea

You can sketch a polynomial from three facts: its zeros say where it meets the x-axis, their multiplicities say whether it crosses or bounces, and the leading term says where the two ends go.

P(x) \approx {\htmlData{sym=0}{a_n}}x^{\htmlData{sym=1}{n}} \quad \text{for large } |x|

What the symbols mean

$a_n$
Leading coefficient
Its sign decides whether the right end goes up or down.
$n$
Degree
Even: both ends point the same way. Odd: opposite ways.

Explanation

Graphing a polynomial is simply drawing its curve, and the good news is you don't need to plot dozens of points to do it. The formula up top says that far from the origin, $P(x)$ behaves like its first term alone, $a_nx^n$ : $a_n$ (the leading coefficient, whose sign says up or down on the right) and $n$ (the degree, which says whether the two ends match). Add in the zeros from last lesson and you've got almost the whole picture. It's a handful of quick checks, not a big computation.

End behavior comes from the leading term alone. Why? For huge $x$ , the highest power grows far faster than everything else. At $x = 100$ , $x^3$ is a million while $5x$ is only 500, so the lower terms barely matter. That gives four cases:

	$a_n > 0$	$a_n < 0$
$n$ even	up on both ends	down on both ends
$n$ odd	down on the left, up on the right	up on the left, down on the right

You can rebuild the table from $x^2$ and $x^3$ : $x^2$ is up on both sides, and $x^3$ goes from down to up. A negative $a_n$ flips it.

At a zero. Think about the factor $(x - r)^m$ near $x = r$ . If $m$ is even, it's a square and can't go negative, so the graph touches the axis and turns back. If $m$ is odd, it changes sign as $x$ passes $r$ , so the graph crosses. Higher multiplicity makes the graph flatter there, since a small number raised to a high power is tiny.

Turning points: at most $n - 1$ .

Sketching. Plot the zeros and the y-intercept, mark which zeros cross and which bounce, draw the ends from the end behavior, and connect everything smoothly. Polynomial graphs never jump or have corners, so there's only one sensible way to join the dots.

See it

p

1.00

q

2.00

Try changing how many times the zeros at −2 and 1 repeat (p and q). Whenever a multiplicity is odd, the graph crosses the axis there; whenever it's even, the graph just touches and turns back. Notice the ends flip too when the total degree changes between even and odd.

Worked examples

Example 1

Sketch $P(x) = (x - 1)^2(x + 3)$ .

1
The zeros are $-3$ , where the graph crosses, and $1$ , which has multiplicity 2, so the graph bounces there.

Practice

Type your answer and check it. Solve them all to complete the lesson.

Problem 1+5 XP

As $x \to \infty$ , what does $P(x) = -2x^3 + 5x - 1$ do?

Problem 2+5 XP

Does $P(x) = (x - 1)^2(x + 3)$ cross or touch the x-axis at $x = 1$ ?

Problem 3+5 XP

Find the y-intercept of $P(x) = (x - 1)^2(x + 3)$ .

Problem 4+5 XP

What is the most turning points a degree 4 polynomial can have?

Problem 5+5 XP

Find the zeros of $P(x) = x^3 - 4x$ .

Problem 6+5 XP

An open box is made from a 12 cm by 12 cm sheet by cutting an $x$ by $x$ square from each corner and folding up the sides, so $V(x) = x(12 - 2x)^2$ . Graph it for $0 < x < 6$ . What is the largest volume, in cm³?

Problem 7+5 XP

As $x \to -\infty$ , what does $P(x) = -x^4 + 3x^3 + 2$ do?

Challenge problem+25 XP

The hardest one here. Get it, and you've got this lesson.

A cubic touches the x-axis at $x = 1$ , crosses it at $x = -3$ , and passes through $(2, -5)$ . Find its y-intercept.

Where it's used

Engineers fit polynomials to sensor data and roller-coaster track profiles; knowing where they turn and how they behave at the ends keeps those fits honest.

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