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Amy Ferguson Moncure

H Algebra 2/Trig · Unit 3

Degree and End Behavior

Name the degree of a polynomial, then say what the ends of the graph do.

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Key Ideas Degree Highest Term Factored Degree End Behavior Standard Form Factored Form x or x²

Degree

The highest power of x.

What to Remember

Degree

The degree is the highest power of x.

If the powers are out of order, find the highest one first.

In factored form, add the exponents. A factor with no power is to the first power.

End behavior

On a graph, read the left side, then the right side.

From an equation, use the degree and the leading coefficient.

Even and positive looks like a quadratic. Even and negative looks like −x².

Odd and positive looks like x³. Odd and negative looks like −x³.

Easy

Degree

The highest power is written first.

Example
State the degree. y = 4x³ − 2x + 7

The degree is the highest power of x.

4x³ is the highest-degree term.

Degree: 3

Answer: Degree: 3

1
State the degree. f(x) = 6x⁴ + 3x² − 1
Show Answer

The degree is the highest power of x.

6x⁴ is the highest-degree term.

Degree: 4

Answer: Degree: 4

2
State the degree. g(x) = −2x⁵ + x² + 4
Show Answer

The degree is the highest power of x.

−2x⁵ is the highest-degree term.

Degree: 5

Answer: Degree: 5

3
State the degree. y = 5x² − 9
Show Answer

The degree is the highest power of x.

5x² is the highest-degree term.

Degree: 2

Answer: Degree: 2

4
State the degree. h(x) = x⁶ − 4x + 8
Show Answer

The degree is the highest power of x.

x⁶ is the highest-degree term.

Degree: 6

Answer: Degree: 6

5
State the degree. p(x) = −7x³ + 2x² − x
Show Answer

The degree is the highest power of x.

−7x³ is the highest-degree term.

Degree: 3

Answer: Degree: 3

Easy

Degree

The polynomial is not in standard form.

Example
State the degree. y = 3x + 6x⁴ − 2

The degree is the highest power of x.

The highest power is not the first term.

6x⁴ is the highest-degree term.

Degree: 4

Answer: Degree: 4

1
State the degree. f(x) = 5 − 2x³ + x
Show Answer

The degree is the highest power of x.

The highest power is not the first term.

−2x³ is the highest-degree term.

Degree: 3

Answer: Degree: 3

2
State the degree. y = 4x² + 7 − 3x⁵
Show Answer

The degree is the highest power of x.

The highest power is not the first term.

−3x⁵ is the highest-degree term.

Degree: 5

Answer: Degree: 5

3
State the degree. g(x) = −6x + 9x² + 1
Show Answer

The degree is the highest power of x.

The highest power is not the first term.

9x² is the highest-degree term.

Degree: 2

Answer: Degree: 2

4
State the degree. h(x) = 2x³ − 8 + 4x⁶
Show Answer

The degree is the highest power of x.

The highest power is not the first term.

4x⁶ is the highest-degree term.

Degree: 6

Answer: Degree: 6

5
State the degree. p(x) = 11x − 4x² + 3x⁴ − 6
Show Answer

The degree is the highest power of x.

The highest power is not the first term.

3x⁴ is the highest-degree term.

Degree: 4

Answer: Degree: 4

Easy

Degree

The polynomial is in factored form.

Example
State the degree. y = (x + 3)(x − 1)²

A factor with no written power is to the first power.

(x + 3)1(x − 1)²

Add the exponents.

1 + 2 = 3

Degree: 3

Answer: Degree: 3

1
State the degree. f(x) = (x − 4)(x + 2)(x − 1)
Show Answer

A factor with no written power is to the first power.

(x − 4)1(x + 2)1(x − 1)1

Add the exponents.

1 + 1 + 1 = 3

Degree: 3

Answer: Degree: 3

2
State the degree. y = −(x + 5)²(x − 3)
Show Answer

A factor with no written power is to the first power.

−(x + 5)²(x − 3)1

Add the exponents.

2 + 1 = 3

Degree: 3

Answer: Degree: 3

3
State the degree. g(x) = (2x − 1)(x + 6)³
Show Answer

A factor with no written power is to the first power.

(2x − 1)1(x + 6)³

Add the exponents.

1 + 3 = 4

Degree: 4

Answer: Degree: 4

4
State the degree. h(x) = (x − 2)⁴(x + 1)²
Show Answer

Every factor already shows its power.

(x − 2)⁴(x + 1)²

Add the exponents.

4 + 2 = 6

Degree: 6

Answer: Degree: 6

5
State the degree. p(x) = 3(x + 4)(x − 7)²
Show Answer

A factor with no written power is to the first power.

3(x + 4)1(x − 7)²

Add the exponents.

1 + 2 = 3

Degree: 3

Answer: Degree: 3

Easy

End Behavior

Read the ends of the graph.

Example
State the end behavior.
−4−3−2−11234−5−4−3−2−112345
−4−3−2−11234−5−4−3−2−112345

x→−∞y→−∞

−4−3−2−11234−5−4−3−2−112345

x→∞y→∞

Answer:
x→−∞y→−∞x→∞y→∞

1
State the end behavior.
−3−2−11234−6−5−4−3−2−11234
Show Answer
−3−2−11234−6−5−4−3−2−11234

x→−∞y→∞

−3−2−11234−6−5−4−3−2−11234

x→∞y→∞

Answer:
x→−∞y→∞x→∞y→∞

2
State the end behavior.
−5−4−3−2−1123456−10−8−6−4−22468101214
Show Answer
−5−4−3−2−1123456−10−8−6−4−22468101214

x→−∞y→−∞

−5−4−3−2−1123456−10−8−6−4−22468101214

x→∞y→∞

Answer:
x→−∞y→−∞x→∞y→∞

3
State the end behavior.
−4−3−2−112345
Show Answer
−4−3−2−112345

x→−∞y→−∞

−4−3−2−112345

x→∞y→−∞

Answer:
x→−∞y→−∞x→∞y→−∞

4
State the end behavior.
−5−4−3−2−11234−8−6−4−22468
Show Answer
−5−4−3−2−11234−8−6−4−22468

x→−∞y→∞

−5−4−3−2−11234−8−6−4−22468

x→∞y→∞

Answer:
x→−∞y→∞x→∞y→∞

5
State the end behavior.
−6−5−4−3−2−11234−16−14−12−10−8−6−4−2246810121416
Show Answer
−6−5−4−3−2−11234−16−14−12−10−8−6−4−2246810121416

x→−∞y→∞

−6−5−4−3−2−11234−16−14−12−10−8−6−4−2246810121416

x→∞y→−∞

Answer:
x→−∞y→∞x→∞y→−∞

Medium

End Behavior

The polynomial is in standard form.

Example
State the end behavior. y = −3x⁴ + x + 6

A positive even degree looks like a quadratic. Both ends go up.

x²

A negative even degree looks like −x². Both ends go down.

−x²

A positive odd degree looks like x³. The left end goes down and the right end goes up.

x³

A negative odd degree looks like −x³. The left end goes up and the right end goes down.

−x³

Use that on this polynomial.

Degree: 4

Leading coefficient: −3

−x²

Even degree and a negative leading coefficient: both ends go down.

x→−∞y→−∞x→∞y→−∞

Answer:
x→−∞y→−∞x→∞y→−∞

1
State the end behavior. f(x) = 2x⁴ − 5x + 1
Show Answer

Degree: 4

Leading coefficient: 2

x²

Even degree and a positive leading coefficient: both ends go up.

x→−∞y→∞x→∞y→∞

Answer:
x→−∞y→∞x→∞y→∞

2
State the end behavior. g(x) = −x⁵ + 4x² − 3
Show Answer

Degree: 5

Leading coefficient: −1

−x³

Odd degree and a negative leading coefficient: the left end goes up and the right end goes down.

x→−∞y→∞x→∞y→−∞

Answer:
x→−∞y→∞x→∞y→−∞

3
State the end behavior. y = 3x³ + 2x − 7
Show Answer

Degree: 3

Leading coefficient: 3

x³

Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.

x→−∞y→−∞x→∞y→∞

Answer:
x→−∞y→−∞x→∞y→∞

4
State the end behavior. h(x) = −4x² + 6x + 1
Show Answer

Degree: 2

Leading coefficient: −4

−x²

Even degree and a negative leading coefficient: both ends go down.

x→−∞y→−∞x→∞y→−∞

Answer:
x→−∞y→−∞x→∞y→−∞

5
State the end behavior. p(x) = 5x⁶ − 2x³ + x
Show Answer

Degree: 6

Leading coefficient: 5

x²

Even degree and a positive leading coefficient: both ends go up.

x→−∞y→∞x→∞y→∞

Answer:
x→−∞y→∞x→∞y→∞

Medium

End Behavior

The polynomial is in factored form.

Example
State the end behavior. y = (x + 6)(x − 2)²

A factor with no written power is to the first power.

(x + 6)1(x − 2)²

Add the exponents.

1 + 2 = 3

Degree: 3

Each factor has leading coefficient 1, and there is no extra number in front.

Leading coefficient: 1

x³

Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.

x→−∞y→−∞x→∞y→∞

Answer:
x→−∞y→−∞x→∞y→∞

1
State the end behavior. f(x) = −(x − 1)(x + 4)(x − 2)
Show Answer

A factor with no written power is to the first power.

−(x − 1)1(x + 4)1(x − 2)1

Add the exponents.

1 + 1 + 1 = 3

Degree: 3

Each factor has leading coefficient 1. The minus in front changes the sign.

Leading coefficient: −1

−x³

Odd degree and a negative leading coefficient: the left end goes up and the right end goes down.

x→−∞y→∞x→∞y→−∞

Answer:
x→−∞y→∞x→∞y→−∞

2
State the end behavior. y = (x + 5)²(x − 1)²
Show Answer

Every factor already shows its power.

(x + 5)²(x − 1)²

Add the exponents.

2 + 2 = 4

Degree: 4

Each factor has leading coefficient 1, and there is no extra number in front.

Leading coefficient: 1

x²

Even degree and a positive leading coefficient: both ends go up.

x→−∞y→∞x→∞y→∞

Answer:
x→−∞y→∞x→∞y→∞

3
State the end behavior. g(x) = (2x + 1)(x − 6)
Show Answer

A factor with no written power is to the first power.

(2x + 1)1(x − 6)1

Add the exponents.

1 + 1 = 2

Degree: 2

The leading part of (2x + 1) is 2x.

Multiply the leading coefficients.

Leading coefficient: 2 × 1 = 2

x²

Even degree and a positive leading coefficient: both ends go up.

x→−∞y→∞x→∞y→∞

Answer:
x→−∞y→∞x→∞y→∞

4
State the end behavior. h(x) = −2(x + 4)³(x − 1)
Show Answer

A factor with no written power is to the first power.

−2(x + 4)³(x − 1)1

Add the exponents.

3 + 1 = 4

Degree: 4

Each factor has leading coefficient 1, so the number in front is the leading coefficient.

Leading coefficient: −2

−x²

Even degree and a negative leading coefficient: both ends go down.

x→−∞y→−∞x→∞y→−∞

Answer:
x→−∞y→−∞x→∞y→−∞

5
State the end behavior. p(x) = (3x − 2)(x + 1)²
Show Answer

A factor with no written power is to the first power.

(3x − 2)1(x + 1)²

Add the exponents.

1 + 2 = 3

Degree: 3

The leading part of (3x − 2) is 3x.

Multiply the leading coefficients.

Leading coefficient: 3 × 1 = 3

x³

Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.

x→−∞y→−∞x→∞y→∞

Answer:
x→−∞y→−∞x→∞y→∞

Hard

End Behavior

One factor is x or x².

Example
State the end behavior. y = x(x − 3)(x + 4)

A factor with no written power is to the first power.

x1(x − 3)1(x + 4)1

Add the exponents.

1 + 1 + 1 = 3

Degree: 3

Each factor has leading coefficient 1, and there is no extra number in front.

Leading coefficient: 1

x³

Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.

x→−∞y→−∞x→∞y→∞

Answer:
x→−∞y→−∞x→∞y→∞

1
State the end behavior. f(x) = −x(x + 2)(x − 5)
Show Answer

A factor with no written power is to the first power.

−x1(x + 2)1(x − 5)1

Add the exponents.

1 + 1 + 1 = 3

Degree: 3

Each factor has leading coefficient 1. The minus in front changes the sign.

Leading coefficient: −1

−x³

Odd degree and a negative leading coefficient: the left end goes up and the right end goes down.

x→−∞y→∞x→∞y→−∞

Answer:
x→−∞y→∞x→∞y→−∞

2
State the end behavior. y = x²(x − 1)(x + 6)
Show Answer

A factor with no written power is to the first power.

x²(x − 1)1(x + 6)1

Add the exponents.

2 + 1 + 1 = 4

Degree: 4

Each factor has leading coefficient 1, and there is no extra number in front.

Leading coefficient: 1

x²

Even degree and a positive leading coefficient: both ends go up.

x→−∞y→∞x→∞y→∞

Answer:
x→−∞y→∞x→∞y→∞

3
State the end behavior. g(x) = −2x²(x + 3)
Show Answer

A factor with no written power is to the first power.

−2x²(x + 3)1

Add the exponents.

2 + 1 = 3

Degree: 3

Each factor has leading coefficient 1, so the number in front is the leading coefficient.

Leading coefficient: −2

−x³

Odd degree and a negative leading coefficient: the left end goes up and the right end goes down.

x→−∞y→∞x→∞y→−∞

Answer:
x→−∞y→∞x→∞y→−∞

4
State the end behavior. h(x) = x(x − 5)²
Show Answer

A factor with no written power is to the first power.

x1(x − 5)²

Add the exponents.

1 + 2 = 3

Degree: 3

Each factor has leading coefficient 1, and there is no extra number in front.

Leading coefficient: 1

x³

Odd degree and a positive leading coefficient: the left end goes down and the right end goes up.

x→−∞y→−∞x→∞y→∞

Answer:
x→−∞y→−∞x→∞y→∞

5
State the end behavior. p(x) = 3x²(2x − 1)(x + 4)
Show Answer

A factor with no written power is to the first power.

3x²(2x − 1)1(x + 4)1

Add the exponents.

2 + 1 + 1 = 4

Degree: 4

The leading part of (2x − 1) is 2x.

Multiply the leading coefficients.

Leading coefficient: 3 × 1 × 2 × 1 = 6

x²

Even degree and a positive leading coefficient: both ends go up.

x→−∞y→∞x→∞y→∞

Answer:
x→−∞y→∞x→∞y→∞

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