Unit 1 · Topic 1.6

Polynomial Functions and End Behavior

As |x| grows, a polynomial behaves like its leading term a_nx^n .

Practice 2: Multiple RepresentationsPractice 3: Communication and Reasoning

Part 1 · Hook

Why this matters

Zoom out on any polynomial graph and the wiggles shrink to nothing; only the ends remain, shooting up or down. The leading term alone decides where they go. The exam asks for that end behavior in limit notation, and readers take the point when the notation is sloppy.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. What is the leading term of 5−3x4+2x5 - 3x^4 + 2x?

  1. −3x4-3x^4
  2. 55
  3. 3x43x^4
  4. 2x2x
Show the answer

The highest power of xx.

  • Correct: −3x4-3x^4:
  • 55:
  • 3x43x^4:
  • 2x2x:

2. ff is an even function. Its graph is symmetric about

  1. the yy-axis
  2. the origin
  3. the xx-axis
  4. the line y=xy = x
Show the answer

f(−x)=f(x)f(-x) = f(x).

  • Correct: the yy-axis:
  • the origin:
  • the xx-axis:
  • the line y=xy = x:

Part 4 · See it

See it first

The graph of p(x) = −0.5x³ + 3x. Its leading term is −0.5x³ (odd degree, negative leading coefficient): the left end rises without bound and the right end falls without bound.
p(x)=−0.5x3+3xp(x) = -0.5x^3 + 3x: odd degree, negative leading coefficient, so limx→−∞p(x)=∞\lim_{x \to -\infty} p(x) = \infty and limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

  1. |x||x| grows largethe leading term anxna_n x^n outgrows every other term
  2. The degree nn is evenboth ends go the same way: up if an>0a_n > 0, down if an<0a_n < 0
  3. The degree nn is oddthe ends go opposite ways: left down and right up if an>0a_n > 0, the reverse if an<0a_n < 0
  4. You report end behaviorwrite both limits in full, such as limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty

Part 6 · Key ideas

Key ideas

  • End behavior: what the outputs do as x→∞x \to \infty and as x→−∞x \to -\infty. For a polynomial, the leading term decides it.
  • Limit notation: limx→∞f(x)=∞\lim_{x \to \infty} f(x) = \infty reads "as xx increases without bound, f(x)f(x) increases without bound."
  • Even degree: ends match. Odd degree: ends opposite. The sign of the leading coefficient picks which way.

Part 7 · Misconception

A common mistake

The wrong idea: If the first few large inputs give negative outputs, then limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty.

What actually happens: End behavior is decided by the leading term, which may take over late. p(x)=2x3−50x2p(x) = 2x^3 - 50x^2 is negative at x=10x = 10 and 2020, but limx→∞p(x)=∞\lim_{x \to \infty} p(x) = \infty because 2x32x^3 wins once x>25x > 25.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

The graph of a polynomial function qq

−4−3−2−112345−6−5−4−3−2−11234xyq

1. The graph shows q(x)q(x)=−0.1(x+3)(x+1)(x−2)(x−4)= -0.1(x + 3)(x + 1)(x - 2)(x - 4). Which describes the right end of the graph?

  1. limx→∞q(x)=∞\lim_{x \to \infty} q(x) = \infty
  2. limx→∞q(x)=−∞\lim_{x \to \infty} q(x) = -\infty
  3. limx→∞q(x)=0\lim_{x \to \infty} q(x) = 0
  4. limx→−∞q(x)=−∞\lim_{x \to -\infty} q(x) = -\infty
Show the answer

The leading term is −0.1x4-0.1x^4: for large xx, x4x^4 is huge and positive, and −0.1-0.1 makes it huge and negative. So limx→∞q(x)=−∞\lim_{x \to \infty} q(x) = -\infty.

  • limx→∞q(x)=∞\lim_{x \to \infty} q(x) = \infty: The right end falls, so the outputs decrease without bound.
  • Correct: limx→∞q(x)=−∞\lim_{x \to \infty} q(x) = -\infty: Right: as xx increases without bound, q(x)q(x) decreases without bound.
  • limx→∞q(x)=0\lim_{x \to \infty} q(x) = 0: The graph does not level off at 00; it keeps falling.
  • limx→−∞q(x)=−∞\lim_{x \to -\infty} q(x) = -\infty: This is about the left end, x→−∞x \to -\infty.

2. The graph shows q(x)q(x)=−0.1(x+3)(x+1)(x−2)(x−4)= -0.1(x + 3)(x + 1)(x - 2)(x - 4). Which describes the left end of the graph?

  1. limx→−∞q(x)=∞\lim_{x \to -\infty} q(x) = \infty
  2. limx→−∞q(x)=−3\lim_{x \to -\infty} q(x) = -3
  3. limx→−∞q(x)=−∞\lim_{x \to -\infty} q(x) = -\infty
  4. limx→∞q(x)=−∞\lim_{x \to \infty} q(x) = -\infty
Show the answer

−0.1x4-0.1x^4 is negative for every large |x||x|, so both ends fall: limx→−∞q(x)=−∞\lim_{x \to -\infty} q(x) = -\infty.

  • limx→−∞q(x)=∞\lim_{x \to -\infty} q(x) = \infty: For even degree, both ends go the same way; here both fall.
  • limx→−∞q(x)=−3\lim_{x \to -\infty} q(x) = -3: −3-3 is a zero; the graph keeps falling past it.
  • Correct: limx→−∞q(x)=−∞\lim_{x \to -\infty} q(x) = -\infty: Right: (−x)4=x4(-x)^4 = x^4 is positive, times −0.1-0.1, so the outputs fall without bound.
  • limx→∞q(x)=−∞\lim_{x \to \infty} q(x) = -\infty: True, but that describes the right end.

3. The graph shows q(x)q(x)=−0.1(x+3)(x+1)(x−2)(x−4)= -0.1(x + 3)(x + 1)(x - 2)(x - 4). What are the degree and the sign of the leading coefficient of qq?

  1. Degree 44, negative
  2. Degree 44, positive
  3. Degree 33, negative
  4. Degree 55, negative
Show the answer

Leading term −0.1⋅x⋅x⋅x⋅x=−0.1x4-0.1 \cdot x \cdot x \cdot x \cdot x = -0.1x^4: even degree (ends match) and negative (both fall).

  • Correct: Degree 44, negative: Right: four factors of xx, and the leading coefficient is −0.1-0.1.
  • Degree 44, positive: A positive leading coefficient with even degree would make both ends rise.
  • Degree 33, negative: Odd degree would send the two ends in opposite directions.
  • Degree 55, negative: There are four factors (x−a)(x - a): degree 44.

Table

Outputs at large inputs

Values of p(x) = 2x^3 - 50x^2 at large inputs x
xx10203050100
p(x)p(x)-3,000-4,0009,000125,0001,500,000

4. The table gives values of p(x)=2x3−50x2p(x) = 2x^3 - 50x^2. Which statement describes the end behavior of pp as xx increases?

  1. limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty
  2. limx→∞p(x)\lim_{x \to \infty} p(x)=1500000= 1500000
  3. limx→∞p(x)=∞\lim_{x \to \infty} p(x) = \infty
  4. limx→∞p(x)=2\lim_{x \to \infty} p(x) = 2
Show the answer

For large xx the leading term 2x32x^3 is far bigger than 50x250x^2, so p(x)p(x) grows without bound: limx→∞p(x)=∞\lim_{x \to \infty} p(x) = \infty.

  • limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty: The first values are negative, but by x=30x = 30 the outputs grow without bound.
  • limx→∞p(x)\lim_{x \to \infty} p(x)=1500000= 1500000: 1,500,0001{,}500{,}000 is one output; the outputs keep growing.
  • Correct: limx→∞p(x)=∞\lim_{x \to \infty} p(x) = \infty: Right: the leading term 2x32x^3 wins for large xx.
  • limx→∞p(x)=2\lim_{x \to \infty} p(x) = 2: 22 is the leading coefficient, not a limit of the outputs.

5. The table gives values of p(x)=2x3−50x2p(x) = 2x^3 - 50x^2. Why are p(10)p(10) and p(20)p(20) negative even though p(x)p(x) grows without bound?

  1. −50x2-50x^2 is larger in size than 2x32x^3 until xx passes 2525.
  2. The table's values were computed incorrectly from the formula.
  3. So the end behavior is really limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty.
  4. Negative outputs mean the leading coefficient is negative.
Show the answer

End behavior is about large inputs. p(x)=2x2(x−25)p(x) = 2x^2(x - 25) is negative for 0<x<250 < x < 25 and positive after, so the leading term takes over only past x=25x = 25.

  • Correct: −50x2-50x^2 is larger in size than 2x32x^3 until xx passes 2525.: Right: 2x3>50x22x^3 > 50x^2 exactly when x>25x > 25; after that the leading term dominates.
  • The table's values were computed incorrectly from the formula.: 2(1000)−50(100)2(1000) - 50(100)=−3000= -3000 is correct.
  • So the end behavior is really limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty.: The later values grow without bound; the end behavior is about large xx.
  • Negative outputs mean the leading coefficient is negative.: The leading coefficient, 22, is positive.

6. Which statement about limx→∞f(x)\lim_{x \to \infty} f(x) for f(x)=x2f(x) = x^2 is written correctly?

  1. f→∞f \to \infty
  2. limx→∞f(x)=∞\lim_{x \to \infty} f(x) = \infty
  3. f(∞)=∞f(\infty) = \infty
  4. limx→∞f(x)→∞\lim_{x \to \infty} f(x) \to \infty
Show the answer

Write limit notation in full: limx→∞f(x)=∞\lim_{x \to \infty} f(x) = \infty. Graders do not accept "f→∞f \to \infty" or f(∞)f(\infty).

  • f→∞f \to \infty: It leaves out the input: say x→∞x \to \infty, in limit notation.
  • Correct: limx→∞f(x)=∞\lim_{x \to \infty} f(x) = \infty: Right: function, input, where the input goes, and where the outputs go.
  • f(∞)=∞f(\infty) = \infty: ∞\infty is not an input; f(∞)f(\infty) means nothing.
  • limx→∞f(x)→∞\lim_{x \to \infty} f(x) \to \infty: Do not mix an arrow into the statement: the limit equals ∞\infty.

Part 9 · Summary

Summary

As |x||x| grows, a polynomial behaves like its leading term anxna_n x^n. Even degree sends both ends the same way and odd degree sends them opposite ways; the sign of ana_n picks the direction. State end behavior in full limit notation, for example limx→−∞p(x)=∞\lim_{x \to -\infty} p(x) = \infty and limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty.

Part 10 · Up next

What comes next

Part 11 · Connections

Connections

Question set for this topic