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Polynomial Functions
and End Behavior

Master the art of graphing polynomial functions, identifying zeros from factorizations, and predicting end behavior from degree and leading coefficient.

Algebra 2

What Is a Polynomial Function?

A polynomial function is a function of the form:
$$ f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 $$
where \( n \) is a non-negative integer and the coefficients \( a_n, a_{n-1}, \ldots, a_0 \) are real numbers with \( a_n \neq 0 \).

Degree
The highest power of \( x \). Determines the shape and number of possible turning points.
Leading Coefficient
The coefficient \( a_n \) of the highest-degree term. Controls the direction the graph opens.

Zeros and Factored Form

A zero (or root) of \( f(x) \) is any value \( c \) such that \( f(c) = 0 \). Graphically, zeros are the x-intercepts where the curve crosses or touches the x-axis.

Odd Multiplicity
Graph crosses the x-axis at the zero (e.g., single root, triple root).
Even Multiplicity
Graph touches (bounces off) the x-axis at the zero (e.g., double root).

Why End Behavior Matters

End behavior describes what happens to \( f(x) \) as \( x \to +\infty \) and \( x \to -\infty \). It gives you the big picture of the graph: does it rise or fall on each side?

\( x \to -\infty, f(x) \to -\infty \)
\( x \to +\infty, f(x) \to +\infty \)
Odd degree, positive leading coefficient

The End Behavior Rule

End behavior depends on two things only: the degree \( n \) and the leading coefficient \( a_n \).

\( a_n > 0 \) \( a_n < 0 \)
Even degree Both ends rise (up, up) Both ends fall (down, down)
Odd degree Left falls, right rises Left rises, right falls

Visualizing All Four Cases

Even degree, \( a_n > 0 \)
 ↑ ↑
Even degree, \( a_n < 0 \)
 ↓ ↓
Odd degree, \( a_n > 0 \)
 ↓ ↑
Odd degree, \( a_n < 0 \)
 ↑ ↓

Degree vs. Leading Coefficient

Both the degree and the leading coefficient shape the graph, but they control different things. Confusing their roles is one of the most common errors.

Feature Degree \( n \) Leading Coeff. \( a_n \)
Controls Overall shape (ends same or opposite) Direction (up or down)
Max turning points \( n - 1 \) No effect
Max real zeros \( n \) No effect
End behavior Even: same direction; Odd: opposite Positive: right rises; Negative: right falls

How to Graph a Polynomial

1
Identify the degree and leading coefficient. This immediately tells you the end behavior.
2
Find the zeros. Factor the polynomial (if possible) and set each factor equal to zero.
3
Determine multiplicity. Each zero's multiplicity tells you if the graph crosses or bounces.
4
Find the y-intercept. Evaluate \( f(0) \) to get the point where the graph crosses the y-axis.
5
Plot and connect. Use end behavior, zeros, and the y-intercept to sketch a smooth, continuous curve.

Example 1: \( f(x) = x^3 - 4x \)

x y -3 -2 -1 1 2 3 -2 0 2

Example 2: \( g(x) = -(x+1)^2(x-3) \)

x y -2 -1 1 2 3 4 (0, 3) -1 3 bounce cross

Example 3: \( h(x) = x^4 - 5x^2 + 4 \)

x y -2 -1 1 2 (0, 4)

How Multiplicity Affects the Graph

Multiplicity 1
Graph crosses the x-axis at a steep angle. The curve passes straight through.
Multiplicity 2
Graph touches and bounces off the x-axis like a parabola would at its vertex.
Multiplicity 3
Graph crosses with an S-shaped flattening (inflection point) at the zero.

Turning Points and Degree

A turning point is where the graph changes from increasing to decreasing (or vice versa). A polynomial of degree \( n \) has at most \( n - 1 \) turning points.

Degree Max Turning Points Example Shape
2 (quadratic) 1 Parabola: one vertex
3 (cubic) 2 S-curve with one hill and one valley
4 (quartic) 3 W-shape or M-shape
5 (quintic) 4 Extended S with more waves

Common Mistakes to Avoid

Ignoring Multiplicity
Drawing the graph crossing at a double root instead of bouncing. Always check the exponent on each factor.
Wrong End Behavior
Confusing even and odd degree effects. Remember: even = both ends same direction; odd = opposite directions.
Forgetting to Expand
Reading the leading coefficient from the factored form without multiplying out. The sign of \( a_n \) can flip when you distribute a negative.

Summary: Polynomial Graphing Checklist

Step What to Find How It Helps
1 Degree and leading coeff. Determines end behavior
2 Zeros from factors x-intercepts of the graph
3 Multiplicity of each zero Cross vs. bounce at each zero
4 y-intercept: \( f(0) \) Starting anchor point
5 Turning points (\( \le n - 1 \)) Shape between zeros

Question 1 of 5

What is the end behavior of \( f(x) = -3x^4 + 2x^2 - 1 \)?

Question 2 of 5

For \( g(x) = (x - 1)^2(x + 3) \), what happens at \( x = 1 \)?

Question 3 of 5

A polynomial of degree 5 can have at most how many turning points?

Question 4 of 5

What are the zeros of \( p(x) = x(x + 2)(x - 5) \)?

Question 5 of 5

Which graph could represent \( f(x) = x^3 - x \)?