Graphs of Polynomial Functions — Interactive Explorer

Adjust the coefficients of a degree-5 polynomial and watch the graph's shape, roots, and end behavior change

Overview

This interactive app lets you explore polynomial functions of degree up to 5, written in the general form:

\[ P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + k \]

(Note: we call the constant term \(k\) here to avoid confusing it with the function name \(P\).)

By modifying the coefficients \(a, b, c, d, e,\) and \(k\), you can visually analyze how each term affects the graph's shape, intercepts, and end behavior. To investigate a lower degree, set the higher-order coefficients to zero — for example, set \( a = b = c = d = 0 \) for a line, or \( a = b = c = 0 \) for a parabola.

Interactive Polynomial Grapher

Enter real-number coefficients below and click Plot Polynomial to update the graph.

Interactive Tutorials & Explorations

A polynomial \( P(x) \) with real coefficients and degree \( n \) has \( n \) complex zeros (not necessarily all distinct). The real zeros appear directly as x-intercepts on the Cartesian plane. Work through the following explorations using the app above; a summary of the key conclusions follows each section.

A. Real solutions of \(x^n + k = 0\)

  1. Set all coefficients to zero except \( a \) and \( k \) (giving \( ax^5 + k = 0 \)). Count the x-intercepts as you vary \( a \) and \( k \).
  2. Set only \( b \) and \( k \) nonzero (degree 4). Which signs of \( k \) produce x-intercepts, and how many?
  3. Set only \( c \) and \( k \) nonzero (degree 3). Count the x-intercepts.
  4. Set only \( d \) and \( k \) nonzero (degree 2). Observe how the number of intercepts depends on \( k \).
  5. Set only \( e \) and \( k \) nonzero (degree 1). Observe the graph.
Conclusions (reveal after trying)

Odd \(n\) (degrees 1, 3, 5): the equation \( x^n + k = 0 \) has exactly 1 real solution for any real \( k \).

Even \(n\) (degrees 2, 4): the equation \( x^n + k = 0 \) has:

  • 0 real solutions when \( k > 0 \),
  • 1 real solution (\( x = 0 \)) when \( k = 0 \),
  • 2 real solutions (\( x = \pm|k|^{1/n} \)) when \( k < 0 \).

Reason: even powers map both \(x\) and \(-x\) to the same positive value, so \(x^n \ge 0\) for even \(n\) — hence \(x^n = -k\) has real solutions only when \(-k \ge 0\).

B. Even and odd polynomial functions

  1. Set \( a = c = e = 0 \). Examine the graph. Is the resulting function even, odd, or neither?
  2. Set \( b = d = k = 0 \). Examine the graph. Is the function even, odd, or neither?
Conclusions (reveal after trying)

Case 1 — only even powers: \( P(x) = bx^4 + dx^2 + k \). This is an even function (\(P(-x) = P(x)\)), with graph symmetric about the y-axis.

Case 2 — only odd powers: \( P(x) = ax^5 + cx^3 + ex \). This is an odd function (\(P(-x) = -P(x)\)), with graph symmetric about the origin (180° rotational symmetry).

C. Zeros of polynomials

Note: a repeated zero of multiplicity \(m\) is counted \(m\) times.

  1. Set \( a \ne 0 \) (degree 5). Adjust the other coefficients so the graph has 1, then 3, then 5 distinct x-intercepts. Can you also get an even number of distinct intercepts?
  2. Set \( a = 0 \) and \( b \ne 0 \) (degree 4). Adjust the other coefficients so the graph has 0, then 2, then 4 distinct x-intercepts.
Conclusions (reveal after trying)

Counting multiplicity: a degree-\(n\) polynomial has exactly \(n\) complex zeros, and the number of real zeros counting multiplicity has the same parity as \(n\) (odd degree → odd number, even degree → even number).

Counting distinct x-intercepts: the number can be any value from 0 up to \(n\). For example, \( x^2 (x-1)^2 (x+1) \) has degree 5 and 3 distinct x-intercepts (at \(x = 0, 1, -1\)), with real zeros counting multiplicity = 5.

D. Leading coefficient test (end behavior)

  1. Set \( a \) to a positive value (degree 5). Does the graph go up or down as \( x \to +\infty \)? As \( x \to -\infty \)? Now flip the sign of \( a \).
  2. Set \( a = 0 \) and \( b > 0 \) (degree 4). Describe the end behavior on both sides. Repeat with \( b < 0 \).
  3. Set \( a = b = 0 \) and \( c > 0 \) (degree 3). Describe the end behavior.
  4. Repeat for degree 2 (leading coefficient \( d \)) and degree 1 (leading coefficient \( e \)).
Conclusions (reveal after trying)
Degree Leading coeff. \( x \to -\infty \) \( x \to +\infty \)
Evenpositive\( +\infty \)\( +\infty \)
Evennegative\( -\infty \)\( -\infty \)
Oddpositive\( -\infty \)\( +\infty \)
Oddnegative\( +\infty \)\( -\infty \)

Frequently Asked Questions

How many real zeros can a degree-5 polynomial have?

Counting multiplicity, exactly 5 complex zeros, and an odd number of real zeros (1, 3, or 5). The number of distinct real x-intercepts can be 0 to 5 — for example, \( x^2(x-1)^2(x+1) \) has 3 distinct intercepts but 5 real zeros counting multiplicity.

How do you determine the end behavior of a polynomial?

End behavior is set by the degree and leading coefficient. Even degree with positive leading coeff. → both ends up. Even degree with negative → both ends down. Odd degree with positive → left down, right up. Odd degree with negative → left up, right down.

What is an even polynomial function?

An even polynomial has only even-power terms: \( P(-x) = P(x) \). Example: \( x^4 - 3x^2 + 1 \). Its graph is symmetric about the y-axis.

What is an odd polynomial function?

An odd polynomial has only odd-power terms: \( P(-x) = -P(x) \). Example: \( x^5 - 3x^3 + x \). Its graph is symmetric about the origin (180° rotational symmetry).

More Tutorials and Practice

Last updated: October 17, 2026