Unit 1 Topic 1.6

Polynomial Functions and End Behavior

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So far you have studied polynomial graphs up close: their zeros, turning points and bends. This topic zooms out. Far from the origin, every polynomial graph looks like the graph of its leading term, and its ends either rise or fall without bound. That long-run behavior is the end behavior, and the exam wants it written in limit notation.

Why the leading term wins

Take p(x)=2x3−50x2p(x) = 2x^3 - 50x^2. At x=10x = 10, 2x3=2,0002x^3 = 2{,}000 and 50x2=5,00050x^2 = 5{,}000: the second term is bigger and p(10)=−3,000p(10) = -3{,}000. At x=100x = 100, 2x3=2,000,0002x^3 = 2{,}000{,}000 and 50x2=500,00050x^2 = 500{,}000: the leading term has pulled ahead, and it stays ahead for every larger xx. A higher power always outgrows a lower one eventually.

So the end behavior of any polynomial is the end behavior of its leading term anxna_n x^n, which depends on only two things: whether nn is even or odd, and whether ana_n is positive or negative.

an>0a_n > 0an<0a_n < 0
even degreelimx→−∞p(x)=∞\lim_{x \to -\infty} p(x) = \infty, limx→∞p(x)=∞\lim_{x \to \infty} p(x) = \infty (both up)limx→−∞p(x)=−∞\lim_{x \to -\infty} p(x) = -\infty, limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty (both down)
odd degreelimx→−∞p(x)=−∞\lim_{x \to -\infty} p(x) = -\infty, limx→∞p(x)=∞\lim_{x \to \infty} p(x) = \infty (down, up)limx→−∞p(x)=∞\lim_{x \to -\infty} p(x) = \infty, limx→∞p(x)=−∞\lim_{x \to \infty} p(x) = -\infty (up, down)
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.
Figure 1. Odd degree with a negative leading coefficient: up on the left, down on the right. LevlPrep original diagram.

Limit notation

The statement limx→∞f(x)=∞\lim_{x \to \infty} f(x) = \infty reads: "as xx increases without bound, f(x)f(x) increases without bound." Likewise limx→−∞f(x)=−∞\lim_{x \to -\infty} f(x) = -\infty means that as xx decreases without bound, f(x)f(x) decreases without bound. Write all four parts every time: lim\lim, the function with its input, where the input goes, and where the outputs go.

These lose the point on the exam: "f→∞f \to \infty" (no input), "f(∞)=∞f(\infty) = \infty" (∞\infty is not an input), and "limx→∞=∞\lim_{x \to \infty} = \infty" (no function).

Finding the leading term

When the polynomial is a product, multiply the leading parts of the factors; you do not need to expand.

Worked example 1. Find the end behavior of h(x)=x2(3−x)3h(x) = x^2(3 - x)^3.

Leading part of (3−x)(3 - x) is −x-x, so of (3−x)3(3 - x)^3 it is (−x)3=−x3(-x)^3 = -x^3. Leading term: x2⋅(−x3)=−x5x^2 \cdot (-x^3) = -x^5.

Odd degree, negative coefficient: limx→−∞h(x)=∞\lim_{x \to -\infty} h(x) = \infty and limx→∞h(x)=−∞\lim_{x \to \infty} h(x) = -\infty.

Worked example 2. Find the end behavior of q(x)q(x)=−0.1(x+3)(x+1)(x−2)(x−4)= -0.1(x + 3)(x + 1)(x - 2)(x - 4).

Leading term: −0.1x4-0.1x^4. Even degree, negative: limx→−∞q(x)=−∞\lim_{x \to -\infty} q(x) = -\infty and limx→∞q(x)=−∞\lim_{x \to \infty} q(x) = -\infty.

Because both ends fall, qq has a global maximum but no global minimum.

What end behavior tells you

  • An odd-degree polynomial has at least one real zero: one of its ends is above the xx-axis and the other below, and its graph has no breaks, so it must cross.
  • An even-degree polynomial has a global maximum (both ends down) or a global minimum (both ends up).
  • End behavior says nothing about the middle of the graph. A table of a few large-ish inputs can mislead if the leading term has not taken over yet.

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