3.3Power Functions and Polynomial Families
Identify power functions, determine the degree and leading coefficient of polynomials, and describe end behavior using the leading-term test.
Power functions and polynomials are the building blocks of calculus. Understanding end behavior and degree prepares you to analyze limits, derivatives, and integrals of polynomial functions.
Essential Question
How does the degree and leading coefficient of a polynomial determine its overall shape and end behavior?
Lesson Overview
A power function is any function of the form f(x) = kxⁿ where k ≠ 0 and n is a real number. Power functions are the building blocks of polynomial functions, which are sums of power functions with non-negative integer exponents. In this lesson we identify power functions, determine the degree and leading coefficient of a polynomial, and use the leading-term test to describe end behavior.
Power Function
f(x) = kxⁿ
k ≠ 0, n is a real number. Single-term function.
Examples: f(x) = 3x², g(x) = −2x⁵, h(x) = x⁰·⁵
Polynomial Function
f(x) = aₙxⁿ + … + a₁x + a₀
Sum of power functions; exponents are non-negative integers.
Smooth (no corners) and continuous (no breaks).
Worked Examples
Identify whether f(x) = −7x⁴ is a power function. If so, state k and n.
A power function has the form f(x) = kxⁿ with a single term.
f(x) = −7x⁴ has exactly one term: k = −7, n = 4.
Find the degree and leading coefficient of p(x) = 6x³ − 2x⁵ + x − 9.
Rewrite in standard form (descending degree): p(x) = −2x⁵ + 6x³ + x − 9.
The highest exponent is 5, so the degree is 5.
The leading term is −2x⁵, so the leading coefficient is −2.
Describe the end behavior of f(x) = 3x⁶ − 5x² + 1 using the leading-term test.
Leading term: 3x⁶. Degree = 6 (even). Leading coefficient = 3 (positive).
Rule: even degree + positive leading coefficient → both ends rise.
As x → −∞, f(x) → +∞ and as x → +∞, f(x) → +∞.
Describe the end behavior of g(x) = −x⁵ + 4x³ − 2x.
Leading term: −x⁵. Degree = 5 (odd). Leading coefficient = −1 (negative).
Rule: odd degree + negative leading coefficient → up left, down right.
As x → −∞, g(x) → +∞ and as x → +∞, g(x) → −∞.
What is the maximum number of turning points for h(x) = 2x⁴ − x³ + 5x − 7?
The degree of h(x) is 4 (highest exponent).
Maximum turning points = degree − 1 = 4 − 1 = 3.
Guided Practice
Is f(x) = 4x³ a power function? State k and n.
Hint: Check: does it have exactly one term of the form kxⁿ?
Write p(x) = 5 − 3x + x⁴ in standard form and identify its degree and leading coefficient.
Hint: Reorder terms from highest to lowest exponent, then read off the leading term.
Use the leading-term test to describe the end behavior of f(x) = −2x⁸ + 3x − 1.
Hint: Even or odd degree? Positive or negative leading coefficient? Apply the four-case rule.
Describe the end behavior of g(x) = 7x³ − x² + 4.
Hint: Odd degree, positive leading coefficient → which end-behavior case?
A polynomial has degree 7. What is the maximum number of turning points it can have?
Hint: Use the formula: max turning points = n − 1.
Key Vocabulary
Power Function
A function of the form f(x) = kxⁿ where k ≠ 0 and n is a real number. It consists of a single term.
Example: f(x) = 5x³ (k = 5, n = 3)
Polynomial Function
A sum of power functions with non-negative integer exponents. Smooth and continuous — no sharp corners or breaks.
Example: p(x) = 2x⁴ − x² + 3
Degree
The highest exponent in a polynomial when written in standard form. Determines the general shape and end behavior.
Example: p(x) = x⁵ + 3x² has degree 5
Leading Coefficient
The coefficient of the leading term (the term with the highest degree). Its sign determines which direction the ends point.
Example: In −4x³ + x, the leading coefficient is −4
End Behavior
The direction the graph travels as x approaches positive or negative infinity. Determined by the leading term alone.
Example: f(x) = x² → both ends rise (↑↑)
Turning Point
A point where the graph changes from increasing to decreasing or vice versa (local max or min). A degree-n polynomial has at most n − 1 turning points.
Example: f(x) = x³ − 3x has 2 turning points
Check Your Understanding
Interactive Practice — 5 Questions
Which of the following is a power function?
What is the degree and leading coefficient of f(x) = −4x⁵ + 7x² − 3?
Describe the end behavior of f(x) = 2x⁴ − x + 6.
How many turning points can a degree-6 polynomial have at most?
Which end behavior matches f(x) = −3x⁷?
Independent Practice
Independent Practice
Determine whether f(x) = −9x⁶ is a power function. If so, identify k and n.
Write q(x) = 3 + 2x³ − x in standard form and state its degree and leading coefficient.
Describe the end behavior of f(x) = 4x² − 7x + 1 using the leading-term test.
Describe the end behavior of g(x) = −5x³ + 2x² − x + 8.
A polynomial of degree 9 has how many turning points at most?
Common Mistakes
Confusing a polynomial with a power function — e.g., calling f(x) = x² + x³ a power function.
A power function has exactly ONE term: f(x) = kxⁿ. Multiple terms make it a polynomial.
Reading the leading coefficient from the first term written, not the highest-degree term.
Always rewrite in standard form (descending degree) first, then read the leading coefficient.
Applying the odd-degree rule to an even-degree polynomial because the leading coefficient is negative.
Check degree parity first (even/odd), then check the sign of the leading coefficient — use all four cases.
Saying a degree-4 polynomial must have exactly 3 turning points.
It has AT MOST n − 1 turning points. The actual number can be fewer (e.g., 1 or 3 for degree 4).
Math Tips
The leading term does all the work for end behavior — ignore all other terms when applying the leading-term test.
Even-degree power functions are symmetric about the y-axis (like a parabola). Odd-degree ones are symmetric about the origin (like a cubic).
Standard form means descending exponents: highest degree first. This makes it easy to spot the leading term.
Memorize the four end-behavior cases as a 2×2 table: (even/odd) × (positive/negative leading coefficient).
Turning points are where the graph "turns around." A degree-n polynomial can turn at most n − 1 times — use this to check if a graph is possible.