Power functions and end behavior

Warm-up

Three quick questions from earlier lessons.

Discover

Move the slider and watch the graph of $y = x^n$ for different powers $n$ .

The graph of y = x to the power n. For even n both ends rise; for odd n the left end falls and the right end rises.

For $n = 3$ , what happens on the far left of the graph?

It goes down
It goes up, like $x^2$

Key idea

For large $|x|$ , a polynomial behaves like its leading term, the term with the highest power.

For

a x^n

: if

n

is even, both ends go the same way (up if

a > 0

, down if

a < 0

). If

n

is odd, the ends go opposite ways (the right end up if

a > 0

, down if

a < 0

).

Worked example

Describe the end behavior of $p(x) = 3x^2 - 2x^5 + 1$ .

Leading term:

-2x^5

The highest power is x⁵, wherever it is written.

Odd degree

5 is odd: the ends go opposite ways.

As

x \to \infty

,

p(x) \to -\infty

; as

x \to -\infty

,

p(x) \to \infty

The coefficient −2 is negative: the right end goes down.

Your turn, guided

Your turn: $q(x) = 4x - 7x^4$ .

Leading term:

-7x^4

The highest power.

Leading coefficient:

-7

, so both ends go down

Type the leading coefficient.

Practice ★

Describe the end behavior of $f(x) = 7x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 3x^{ 7 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 3x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = 8x^{ 3 }$ .
1. As $x \to \infty$ , $f(x) \to \infty$ ; as $x \to -\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$

Watch out!

Practice ★★

Describe the end behavior of $f(x) = -9x^{ 6 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -8x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$
Describe the end behavior of $f(x) = -6x^{ 4 }$ .
1. As $x \to \pm\infty$ , $f(x) \to -\infty$
2. As $x \to \pm\infty$ , $f(x) \to \infty$
3. As $x \to \infty$ , $f(x) \to -\infty$ ; as $x \to -\infty$ , $f(x) \to \infty$

$p(x) = 7x^2 - 9x^5 + 9$ . What is the leading coefficient of $p$ ?
$p(x) = 9x^2 - 7x^5 + 2$ . What is the leading coefficient of $p$ ?
$p(x) = 3x^2 - 5x^5 + 8$ . What is the leading coefficient of $p$ ?

Exit ticket

Three questions to check you have got it.

$p(x) = 7x^2 - 9x^5 + 9$ . What is the leading coefficient of $p$ ?
$p(x) = 9x^2 - 7x^5 + 2$ . What is the leading coefficient of $p$ ?
$p(x) = 3x^2 - 5x^5 + 8$ . What is the leading coefficient of $p$ ?

Summary card

End behavior

Only the leading term (highest power) matters for large $|x|$ .
Even degree: both ends the same way. Odd degree: opposite ways.
A negative leading coefficient flips the picture.

Watch out: the leading term is not always written first.

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