3.4 Graphs of Polynomial Functions
Zeros, multiplicities, end behavior, and the Intermediate Value Theorem
Understanding how to read and sketch polynomial graphs is essential for modeling real-world phenomena — from the trajectory of a projectile to the profit curve of a business. Recognizing how zeros and their multiplicities shape a graph lets you extract meaningful information without a calculator.
Essential Question
How do the zeros and their multiplicities of a polynomial function determine the shape of its graph?
Lesson Overview
Every polynomial function has a graph that is a smooth, continuous curve with no sharp corners or breaks. The key to sketching these graphs is understanding where the function equals zero, how it behaves near each zero (based on multiplicity), and what happens as x grows very large or very small (end behavior).
Zeros & Factors
If f(c) = 0, then (x − c) is a factor of f(x). The power m of the factor (x − c)^m is called the multiplicity of the zero x = c.
Multiplicity Rules
Odd multiplicity → graph crosses the x-axis at x = c. Even multiplicity → graph bounces (touches but does not cross) at x = c.
Worked Examples
Find all zeros and their multiplicities for f(x) = (x − 2)³(x + 5)²(x − 1).
Set each factor equal to zero.
(x − 2)³ = 0 → x = 2, multiplicity 3 (odd → crosses)
(x + 5)² = 0 → x = −5, multiplicity 2 (even → bounces)
(x − 1)¹ = 0 → x = 1, multiplicity 1 (odd → crosses)
Determine the end behavior of f(x) = −3x⁴ + 7x² − 2.
Identify degree: 4 (even). Leading coefficient: −3 (negative).
Even degree + negative leading coeff → both ends point DOWN.
As x → −∞, f(x) → −∞; as x → +∞, f(x) → −∞.
Sketch the key features of f(x) = x²(x − 3)(x + 2)².
Zeros: x = 0 (mult 2, bounce), x = 3 (mult 1, cross), x = −2 (mult 2, bounce).
Degree = 2 + 1 + 2 = 5 (odd). Leading coeff = +1 (positive).
End behavior: down-left (x → −∞), up-right (x → +∞).
y-intercept: f(0) = 0²(0 − 3)(0 + 2)² = 0.
At most 5 − 1 = 4 turning points.
Use the IVT to show f(x) = x³ − 2x − 5 has a zero on [2, 3].
f(2) = 8 − 4 − 5 = −1 (negative)
f(3) = 27 − 6 − 5 = 16 (positive)
f is continuous (polynomial). f(2) · f(3) = (−1)(16) < 0.
By IVT, there exists c ∈ (2, 3) with f(c) = 0.
How many turning points can f(x) = 2x⁵ − x³ + 4x have at most?
Degree of f(x) = 5.
Maximum turning points = degree − 1 = 5 − 1 = 4.
Guided Practice
List all zeros and their multiplicities for g(x) = (x + 4)²(x − 6)(x + 1)³.
Hint: Set each factor to zero and read the exponent as the multiplicity.
Describe the end behavior of h(x) = 5x⁶ − 3x⁴ + x.
Hint: Focus only on the leading term 5x⁶. What is the degree parity and sign of the leading coefficient?
For f(x) = −x³(x − 2)², does the graph cross or bounce at x = 0? At x = 2?
Hint: Check the multiplicity of each zero: odd → crosses, even → bounces.
Use the IVT to confirm a zero of p(x) = x³ + x − 1 exists on [0, 1].
Hint: Evaluate p(0) and p(1). Do they have opposite signs?
A polynomial has zeros at x = −3 (mult 1), x = 0 (mult 2), and x = 5 (mult 3). What is the minimum possible degree?
Hint: Add up all the multiplicities: 1 + 2 + 3 = ?
Key Vocabulary
Zero (Root)
A value c such that f(c) = 0. Corresponds to an x-intercept of the graph and a factor (x − c) of the polynomial.
Example: f(x) = (x − 3)(x + 1) has zeros at x = 3 and x = −1
Multiplicity
The exponent m on the factor (x − c)^m in the factored form of a polynomial. Determines whether the graph crosses (odd m) or bounces (even m) at x = c.
Example: In (x − 2)³, the zero x = 2 has multiplicity 3 (odd → crosses)
Turning Point
A point where the graph changes from increasing to decreasing (local max) or decreasing to increasing (local min). A degree-n polynomial has at most n − 1 turning points.
Example: f(x) = x³ − 3x has 2 turning points (degree 3 → at most 2)
Intermediate Value Theorem (IVT)
If f is continuous on [a, b] and f(a) · f(b) < 0 (opposite signs), then there exists at least one c in (a, b) where f(c) = 0.
Example: f(1) = −2 and f(2) = 3 → IVT guarantees a zero in (1, 2)
Local Maximum / Minimum
A turning point where the function value is greater (max) or less (min) than nearby values. Distinct from the global (absolute) max or min over all real numbers.
Example: f(x) = −x⁴ + 4x² has local maxima at x = ±√2
Quick Check
Interactive Practice — 5 Questions
For f(x) = (x − 4)²(x + 1), what is the multiplicity of the zero x = 4?
A zero with odd multiplicity means the graph _____ the x-axis at that zero.
What is the maximum number of turning points for a degree-6 polynomial?
For f(x) = −2x³(x + 3)², which end behavior is correct?
The Intermediate Value Theorem guarantees a zero on [a, b] when:
Common Mistakes
Assuming every zero causes the graph to cross the x-axis.
Even-multiplicity zeros cause the graph to bounce (touch and turn back). Only odd-multiplicity zeros produce a crossing.
Confusing the degree of the polynomial with the number of zeros.
A degree-n polynomial has exactly n zeros (counting multiplicity, in ℂ). The number of real x-intercepts can be fewer.
Ignoring the sign of the leading coefficient when determining end behavior.
Both degree parity AND the sign of the leading coefficient determine end behavior. A negative leading coefficient flips the direction.
Applying IVT without checking that f is continuous on the interval.
IVT requires continuity. Polynomials are always continuous, but rational functions may have discontinuities — check first.
Math Tips
The sum of all multiplicities equals the degree of the polynomial.
End behavior is determined entirely by the leading term aₙxⁿ — all other terms become negligible for large |x|.
To find the y-intercept quickly, evaluate f(0): substitute x = 0 into the factored form.
A graph that bounces at a zero looks like a parabola tangent to the x-axis at that point.
Sketch in order: (1) zeros + multiplicities, (2) end behavior, (3) y-intercept, (4) smooth curve connecting all features.