Graphs of Polynomials End Behavior

Let’s delve into the end behaviour of polynomial graphs! This describes what happens to the graph of a polynomial function as the x values become very large (approaching positive infinity, x → ∞) or very small (approaching negative infinity, x → -∞).

The Role of the Leading Term 🚀

The end behaviour of a polynomial is determined solely by its leading term. The leading term is the term with the highest degree (highest exponent) in the polynomial. For very large or very small values of x, this term will dominate and essentially dictate the overall direction of the graph, making all other terms insignificant in comparison.

Let’s consider a polynomial function f(x) in the standard form: f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀

Here, aₙxⁿ is the leading term, where:

• n is the degree of the polynomial (the highest exponent).
• aₙ is the leading coefficient (the coefficient of the highest-degree term).

The end behaviour depends on two key characteristics of this leading term:

1. Whether the degree n is even or odd.
2. Whether the leading coefficient aₙ is positive or negative.

Understanding the Four Scenarios 🧭

Let’s break down the end behaviour into four distinct cases based on these two characteristics:

1. Even Degree (n is even) 짝수 차수

If the highest exponent n is an even number (e.g., 2, 4, 6, etc.), the ends of the graph will point in the same direction. This is similar to a quadratic function like y = x² or y = -x².

• Case 1: Even Degree, Positive Leading Coefficient (aₙ > 0) ⬆️⬆️

  • As x approaches positive infinity (x → ∞), f(x) approaches positive infinity (f(x) → ∞).
  • As x approaches negative infinity (x → -∞), f(x) also approaches positive infinity (f(x) → ∞).
  • Visual Aid: Both ends of the graph go up (like a smiley face parabola 😊).

• Case 2: Even Degree, Negative Leading Coefficient (aₙ < 0) ⬇️⬇️

  • As x approaches positive infinity (x → ∞), f(x) approaches negative infinity (f(x) → -∞).
  • As x approaches negative infinity (x → -∞), f(x) also approaches negative infinity (f(x) → -∞).
  • Visual Aid: Both ends of the graph go down (like a frowny face parabola ☹️).

2. Odd Degree (n is odd) 홀수 차수

If the highest exponent n is an odd number (e.g., 1, 3, 5, etc.), the ends of the graph will point in opposite directions. This is similar to a linear function like y = x or a cubic function like y = x³.

• Case 3: Odd Degree, Positive Leading Coefficient (aₙ > 0) ⬇️⬆️

  • As x approaches positive infinity (x → ∞), f(x) approaches positive infinity (f(x) → ∞).
  • As x approaches negative infinity (x → -∞), f(x) approaches negative infinity (f(x) → -∞).
  • Visual Aid: The graph goes down on the left and up on the right (like a rising slide 🎢).

• Case 4: Odd Degree, Negative Leading Coefficient (aₙ < 0) ⬆️⬇️

  • As x approaches positive infinity (x → ∞), f(x) approaches negative infinity (f(x) → -∞).
  • As x approaches negative infinity (x → -∞), f(x) approaches positive infinity (f(x) → ∞).
  • Visual Aid: The graph goes up on the left and down on the right (like a falling slide 📉).

Summary Table 📊

This table provides a concise overview of how to determine the end behaviour of a polynomial function.

Practice Questions 📝

Question 1: Describe the End Behaviour

For each polynomial function, describe its end behaviour using the x → ±∞ and f(x) → ±∞ notation, and include an emoji visual aid.

(a) f(x) = 3x⁴ - 2x² + 5 (b) g(x) = -x³ + 7x - 1 (c) h(x) = -2x⁶ + 8x⁵ - 10x (d) k(x) = 0.5x⁵ - x⁴ + 3x + 9

Question 2: Infer from End Behaviour

A polynomial graph shows the following end behaviour:

• As x → ∞, f(x) → -∞.
• As x → -∞, f(x) → ∞.

(a) Is the degree of this polynomial even or odd? (b) Is its leading coefficient positive or negative?

Question 3: Match End Behaviour to Polynomial Form

Match each end behaviour description to the characteristic of its leading term:

Solutions to Practice Questions ✅

Solution 1:

(a) f(x) = 3x⁴ - 2x² + 5

• Leading Term: 3x⁴
• Degree: 4 (Even)
• Leading Coefficient: 3 (Positive)
• End Behaviour: As x → ∞, f(x) → ∞. As x → -∞, f(x) → ∞. ⬆️⬆️

(b) g(x) = -x³ + 7x - 1

• Leading Term: -x³
• Degree: 3 (Odd)
• Leading Coefficient: -1 (Negative)
• End Behaviour: As x → ∞, f(x) → -∞. As x → -∞, f(x) → ∞. ⬆️⬇️

(c) h(x) = -2x⁶ + 8x⁵ - 10x

• Leading Term: -2x⁶
• Degree: 6 (Even)
• Leading Coefficient: -2 (Negative)
• End Behaviour: As x → ∞, f(x) → -∞. As x → -∞, f(x) → -∞. ⬇️⬇️

(d) k(x) = 0.5x⁵ - x⁴ + 3x + 9

• Leading Term: 0.5x⁵
• Degree: 5 (Odd)
• Leading Coefficient: 0.5 (Positive)
• End Behaviour: As x → ∞, f(x) → ∞. As x → -∞, f(x) → -∞. ⬇️⬆️

Solution 2:

The polynomial graph shows:

• As x → ∞, f(x) → -∞.
• As x → -∞, f(x) → ∞. This matches the ⬆️⬇️ visual aid.

(a) Is the degree of this polynomial even or odd?

• Since the ends point in opposite directions (one up, one down), the degree must be odd.

(b) Is its leading coefficient positive or negative?

• For an odd-degree polynomial, if the left end goes up and the right end goes down, the leading coefficient is negative.

Solution 3:

1. Both ends go up ⬆️⬆️: (C) Even degree, positive coefficient.
2. Left end down, right end up ⬇️⬆️: (A) Odd degree, positive coefficient.
3. Both ends go down ⬇️⬇️: (B) Even degree, negative coefficient.
4. Left end up, right end down ⬆️⬇️: (D) Odd degree, negative coefficient.

Understanding these concepts of end behaviour, along with the behavior at x-intercepts, significantly helps in sketching and interpreting polynomial graphs.