Graphing Polynomial Functions

Understanding polynomial behavior and graphs

Graphing Polynomial Functions

End Behavior

Determined by the leading term anxna_nx^n:

Odd degree:

  • an>0a_n > 0: falls left, rises right ↙↗
  • an<0a_n < 0: rises left, falls right ↖↘

Even degree:

  • an>0a_n > 0: rises both sides ↗↗
  • an<0a_n < 0: falls both sides ↘↘

Zeros and Multiplicity

Zero: value where f(x)=0f(x) = 0 (x-intercept)

Multiplicity: how many times the factor appears

Even multiplicity: graph touches x-axis and bounces Odd multiplicity: graph crosses x-axis

Example: f(x)=(x−2)2(x+1)f(x) = (x - 2)^2(x + 1)

  • Zero at x=2x = 2 (multiplicity 2, bounces)
  • Zero at x=−1x = -1 (multiplicity 1, crosses)

Turning Points

A polynomial of degree nn has at most n−1n - 1 turning points.

Y-Intercept

Evaluate f(0)f(0) to find where graph crosses y-axis.

Key Features

  1. Degree determines end behavior
  2. Leading coefficient affects direction
  3. Zeros show x-intercepts
  4. Multiplicity affects crossing behavior

📚 Practice Problems

1Problem 1easy

❓ Question:

Describe the end behavior of f(x)=−2x4+3x2−1f(x) = -2x^4 + 3x^2 - 1

💡 Show Solution

Leading term: −2x4-2x^4

Degree: 4 (even) Leading coefficient: -2 (negative)

For even degree with negative leading coefficient:

  • Left end: falls (goes to −∞-\infty)
  • Right end: falls (goes to −∞-\infty)

Answer: Falls on both ends ↘↘

2Problem 2medium

❓ Question:

Find all zeros and their multiplicities: f(x)=x3(x−2)2(x+1)f(x) = x^3(x - 2)^2(x + 1)

💡 Show Solution

Set each factor equal to zero:

From x3x^3:

  • Zero at x=0x = 0, multiplicity 3 (odd, crosses)

From (x−2)2(x - 2)^2:

  • Zero at x=2x = 2, multiplicity 2 (even, bounces)

From (x+1)(x + 1):

  • Zero at x=−1x = -1, multiplicity 1 (odd, crosses)

Answer:

  • x=0x = 0 (mult. 3, crosses)
  • x=2x = 2 (mult. 2, bounces)
  • x=−1x = -1 (mult. 1, crosses)

3Problem 3medium

❓ Question:

What is the maximum number of turning points for f(x)=5x6−3x4+x2−7f(x) = 5x^6 - 3x^4 + x^2 - 7?

💡 Show Solution

The degree of the polynomial is 6.

A polynomial of degree nn has at most n−1n - 1 turning points.

Max turning points=6−1=5\text{Max turning points} = 6 - 1 = 5

Answer: Maximum of 5 turning points

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