End Behavior Anchor Chart End Behavior Breakdown
The Leading Coefficient Test & Limit Notation
Pre-Calculus Reference
Even Degree / Positive LC
Example: \( f(x) = 3x^4 - 5x + 2 \)
Limit Notation:
\( \lim_{x \to \infty} f(x) = \infty \)
\( \lim_{x \to -\infty} f(x) = \infty \)
"Up and Up"
Even Degree / Negative LC
Example: \( f(x) = -2x^2 + x + 7 \)
Limit Notation:
\( \lim_{x \to \infty} f(x) = -\infty \)
\( \lim_{x \to -\infty} f(x) = -\infty \)
"Down and Down"
Odd Degree / Positive LC
Example: \( f(x) = x^3 + 2x^2 \)
Limit Notation:
\( \lim_{x \to \infty} f(x) = \infty \)
\( \lim_{x \to -\infty} f(x) = -\infty \)
"Down and Up"
Odd Degree / Negative LC
Example: \( f(x) = -4x^5 + 2x \)
Limit Notation:
\( \lim_{x \to \infty} f(x) = -\infty \)
\( \lim_{x \to -\infty} f(x) = \infty \)
"Up and Down"
Pro Tip: Always identify the highest degree term first. Don't be fooled by the order of the terms—polynomials aren't always written in standard form!
Polynomial Pulse Slides Polynomial Pulse
Analyzing End Behavior & Graph Trends
\( f(x) = a_n x^n + ... \) | \( \lim_{x \to \infty} f(x) \)
Warm-up: Dance Moves
5 Minutes
When the teacher calls out a combination, use your arms to model the end behavior of the graph.
The Combinations
Positive Even
Negative Even
Positive Odd
Negative Odd
Pos Odd
Left down, Right up
Pos Even
Both arms up
Video Analysis
Polynomial Graphs Challenge (6:53 - 7:30)
5 Minutes
Embedded media
Watch for:
Identifying the degree
Leading coefficient sign
Formal Limit Notation
"We know the graph really goes for infinity in both directions..."
The Calculus Language
\( \lim_{x \to \infty} f(x) = \infty \)
"As \( x \) approaches positive infinity (moves to the right), the function \( f(x) \) approaches infinity (goes up)."
\( \lim_{x \to -\infty} f(x) = -\infty \)
"As \( x \) approaches negative infinity (moves to the left), the function \( f(x) \) approaches negative infinity (goes down)."
Translate the "Dance" to "Limits"!
End Behavior Sort
20 Minutes
Work in groups to categorize 20 unique polynomial functions into the four end-behavior buckets.
Bucket 1: Up / Up
Even Degree, Pos LC
Bucket 2: Down / Down
Even Degree, Neg LC
Bucket 3: Down / Up
Odd Degree, Pos LC
Bucket 4: Up / Down
Odd Degree, Neg LC
The Rules
Identify Degree
Identify LC Sign
Place Card
Record limit notation for 5 chosen cards
Exit Ticket
Identify the end behavior and write the formal limit notation for:
\( g(x) = -5x^7 + 3x^4 - 2x + 11 \)
\( \lim_{x \to \infty} g(x) = \_\_\_ \)
\( \lim_{x \to -\infty} g(x) = \_\_\_ \)
End Behavior Sort Activity End Behavior Sort
Pre-Calculus Group Activity
Name:
Date:
Instructions
Work in groups to analyze the 20 polynomial functions provided on the cards.
Identify the highest degree and the sign of the leading coefficient for each.
Sort each function into one of the four categories on your recording sheet.
For each category, choose one function and write its formal limit notation in the space provided.
CARD 1
\( f(x) = 2x^4 - 3x + 1 \)
CARD 2
\( g(x) = -x^3 + 5x \)
CARD 3
\( h(x) = x^5 - 2x^2 + x \)
CARD 4
\( p(x) = -3x^6 + 4x^3 \)
CARD 5
\( f(x) = (x-2)(x+3)(x-1) \)
CARD 6
\( g(x) = -2(x^2+1)(x-4) \)
CARD 7
\( h(x) = 4 - 2x + 7x^2 \)
CARD 8
\( p(x) = 10x - x^5 \)
CARD 9
\( f(x) = \frac{1}{2}x^3 + 4 \)
CARD 10
\( g(x) = -(x+1)^2(x-2)^2 \)
CARD 11
\( h(x) = x(x^2-9) \)
CARD 12
\( p(x) = -5x^4 + 3x^2 - x \)
CARD 13
\( f(x) = 15x^2 - x^4 \)
CARD 14
\( g(x) = (2x+1)^2 \)
CARD 15
\( h(x) = -8x^3 - 4x + 2 \)
CARD 16
\( p(x) = x^6 + x^4 + x^2 \)
CARD 17
\( f(x) = -x(x-1)(x+1) \)
CARD 18
\( g(x) = (x+5)^2(x-1) \)
CARD 19
\( h(x) = 5x^5 - 100 \)
CARD 20
\( p(x) = -2x^8 + 3x^7 \)
Group Recording Sheet
UP / UP (Even Degree, Pos LC)
Card numbers:
Limit Notation Example:
\( \lim_{x \to \infty} f(x) = \) ________
\( \lim_{x \to -\infty} f(x) = \) ________
DOWN / DOWN (Even Degree, Neg LC)
Card numbers:
Limit Notation Example:
\( \lim_{x \to \infty} f(x) = \) ________
\( \lim_{x \to -\infty} f(x) = \) ________
DOWN / UP (Odd Degree, Pos LC)
Card numbers:
Limit Notation Example:
\( \lim_{x \to \infty} f(x) = \) ________
\( \lim_{x \to -\infty} f(x) = \) ________
UP / DOWN (Odd Degree, Neg LC)
End Behavior Answer Key Teacher Answer Key
End Behavior Sort & Exit Ticket
End Behavior Sort Key
Bucket 1: Up / Up (Even Degree, Pos LC)
Card 1: \( 2x^4... \) Deg 4, LC +2
Card 7: \( 7x^2... \) Deg 2, LC +7
Card 14: \( (2x+1)^2 \) Deg 2, LC +4
Card 16: \( x^6... \) Deg 6, LC +1
Bucket 2: Down / Down (Even Degree, Neg LC)
Card 4: \( -3x^6... \) Deg 6, LC -3
Card 10: \( -(x+1)^2(x-2)^2 \) Deg 4, LC -1
Card 12: \( -5x^4... \) Deg 4, LC -5
Card 13: \( -x^4... \) Deg 4, LC -1
Card 20: \( -2x^8... \) Deg 8, LC -2
Bucket 3: Down / Up (Odd Degree, Pos LC)
Card 3: \( x^5... \) Deg 5, LC +1
Card 5: \( (x-2)(x+3)(x-1) \) Deg 3, LC +1
Card 9: \( \frac{1}{2}x^3... \) Deg 3, LC +1/2
Card 11: \( x(x^2-9) \) Deg 3, LC +1
Card 18: \( (x+5)^2(x-1) \) Deg 3, LC +1
Card 19: \( 5x^5... \) Deg 5, LC +5
Bucket 4: Up / Down (Odd Degree, Neg LC)
Card 2: \( -x^3... \) Deg 3, LC -1
Card 6: \( -2(x^2+1)(x-4) \) Deg 3, LC -2
Card 8: \( -x^5... \) Deg 5, LC -1
Card 15: \( -8x^3... \) Deg 3, LC -8
Card 17: \( -x(x-1)(x+1) \) Deg 3, LC -1
Exit Ticket Solution
Function: \( g(x) = -5x^7 + 3x^4 - 2x + 11 \)
Analysis
Degree: 7 (Odd)
LC: -5 (Negative)
End Behavior
Starts HIGH, Ends LOW
Formal Limit Notation
\( \lim_{x \to \infty} g(x) = -\infty \)
\( \lim_{x \to -\infty} g(x) = \infty \)
