Math Core

You now know how to find the zeros of a polynomial. In this lesson you put that knowledge on the coordinate plane. With just the factored form and the leading term, you can sketch a polynomial's graph accurately: where it crosses the $x$ -axis, where it bounces off, and what it does far to the left and right.

End behavior

Far from the origin, the leading term $a x^n$ is so much larger than every other term that it controls the whole graph. For example, at $x = 100$ the terms of $x^3 - 4x$ are $1{,}000{,}000$ and $-400$ ; the second barely matters. So the end behavior of a polynomial, what $y$ does as $x \to \infty$ and $x \to -\infty$ , depends only on the degree $n$ and the sign of the leading coefficient $a$ .

End behavior from the leading term

	$a > 0$	$a < 0$
$n$ even	up on both sides	down on both sides
$n$ odd	down on the left, up on the right	up on the left, down on the right

Even degree behaves like $y = x^2$ (both ends point the same way). Odd degree behaves like $y = x^3$ (the ends point opposite ways). A negative leading coefficient flips the picture upside down.

The quartic y = x⁴ − 4x² (even degree, positive leading coefficient) rises on both ends. The cubic y = x³ − 4x (odd degree, positive leading coefficient) falls on the left and rises on the right.Open in grapher →

In limit notation, "down on the left, up on the right" is written: as $x \to -\infty$ , $f(x) \to -\infty$ , and as $x \to \infty$ , $f(x) \to \infty$ .

Zeros and multiplicity

Each real zero is an $x$ -intercept. The multiplicity of the zero tells you how the graph behaves there.

Behavior at a zero

Odd multiplicity (1, 3, …): the graph crosses the $x$ -axis. At multiplicity $1$ it crosses like a line; at multiplicity $3$ it flattens out as it crosses, like $y = x^3$ at the origin.
Even multiplicity (2, 4, …): the graph touches the $x$ -axis and turns back, like $y = x^2$ at its vertex.

The reason is the sign of the factor. Near $x = r$ , the factor $(x - r)^k$ changes sign as $x$ passes $r$ when $k$ is odd, but stays positive on both sides when $k$ is even. The rest of the factors barely change near $r$ , so $f(x)$ changes sign only at odd-multiplicity zeros.

Non-real zeros don't produce $x$ -intercepts at all. A factor like $x^2 + 4$ is always positive, so it affects the shape but never touches the axis.

Turning points

A turning point is where the graph changes from rising to falling or from falling to rising (a local maximum or minimum).

Turning points

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

A cubic can have $2$ turning points or none (like $y = x^3$ ). A quartic can have $3$ or $1$ . If a graph has $4$ turning points, its degree is at least $5$ .

Putting it together

To sketch a polynomial in factored form:

Find the end behavior from the degree and the leading coefficient.
Find the zeros and their multiplicities: cross or touch?
Find the $y$ -intercept by computing $f(0)$ .
Draw a smooth curve that starts with the left-end behavior, passes through the intercepts in order, and finishes with the right-end behavior.

Worked example: A cubic with three simple zeros

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

End behavior: multiplying the $x$ 's gives the leading term $x^3$ . Odd degree, positive leading coefficient: down on the left, up on the right.
Zeros: $-2$ , $1$ and $3$ , each with multiplicity $1$ , so the graph crosses at all three.
$y$ -intercept: $f(0) = (2)(-1)(-3) = 6$ .

Starting low on the left, the graph rises to cross at $x = -2$ , turns, and comes back down through $(0, 6)$ to cross at $x = 1$ , dips below the axis, then turns up to cross at $x = 3$ and rises forever. That's $2$ turning points, the most a cubic can have.

f(x) = (x + 2)(x − 1)(x − 3) crosses the x-axis at −2, 1 and 3 and has y-intercept 6.Open in grapher →

Worked example: A double zero and a negative leading coefficient

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

End behavior: the leading term is $-x^3$ . Odd degree, negative: up on the left, down on the right.
Zeros: $-1$ with multiplicity $2$ (touch and turn) and $2$ with multiplicity $1$ (cross).
$y$ -intercept: $g(0) = -(1)^2(-2) = 2$ .

The graph comes down from the upper left, touches the axis at $x = -1$ and turns back up, passes through $(0, 2)$ , turns again, and falls through the axis at $x = 2$ .

g(x) = −(x + 1)²(x − 2) touches the x-axis at −1 (double zero) and crosses at 2.Open in grapher →

Common mistake

Don't read the end behavior from the first factor you see. For $g(x) = -(x + 1)^2(x - 2)$ , the leading term comes from multiplying the leading terms of all the factors, including the negative sign out front: $-1 \cdot x^2 \cdot x = -x^3$ .

From a graph to an equation

The process runs backward, too. Read the zeros and their behavior off the graph, write the factors, and use one more point to find the leading coefficient $a$ .

Worked example: Writing the equation of a graph

A cubic touches the $x$ -axis at $x = -3$ , crosses it at $x = 1$ , and passes through $(0, -9)$ . Find its equation.

Touching at $-3$ means even multiplicity; since the degree is $3$ , the zeros are $-3$ (multiplicity $2$ ) and $1$ (multiplicity $1$ ):

f(x) = a(x + 3)^2(x - 1).

Use the $y$ -intercept: $f(0) = a(9)(-1) = -9a$ . Setting $-9a = -9$ gives $a = 1$ , so

f(x) = (x + 3)^2(x - 1).

f(x) = (x + 3)²(x − 1) touches the x-axis at −3 and crosses at 1.Open in grapher →

Tip

Use the $y$ -intercept as a check on any sketch: if your curve crosses the $y$ -axis above the origin, $f(0)$ had better be positive. A wrong sign here usually means a sign error in a factor or in the leading coefficient.

Practice

Practice 1

Describe the end behavior of $f(x) = -2x^5 + x^2 - 3$ .

Practice 2

How does the graph of $f(x) = (x - 4)^2(x + 1)$ behave at $x = 4$ ?

Practice 3

What is the greatest number of turning points the graph of a degree- $6$ polynomial can have?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

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

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Find the $x$ -intercepts of $f(x) = x^3 - x^2 - 6x$ .

Separate answers with commas, e.g. 2, -5

Practice 6

Which function has this graph?

y = (x + 3)*(x - 1)^2(-3, 0)(1, 0)(0, 3)Open in grapher →

Practice 7

A cubic has zeros $-1$ , $2$ and $4$ and a $y$ -intercept of $16$ . It can be written $f(x) = a(x + 1)(x - 2)(x - 4)$ . Find $a$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

For which values of $x$ is $f(x) = x^4 - 4x^3$ negative?

Enter an inequality, e.g. x >= 4, -2 < x <= 3, or x < 1 or x > 5