Lesson 1: Graphing Polynomial Functions

Property

A polynomial function has the form

where $a_0, a_1, a_2, \ldots, a_n$ are constants and $a_n \neq 0$. The coefficient $a_n$ of the highest power term is called the lead coefficient. Polynomials can be written in descending powers, where terms are ordered from the highest degree to the lowest, or in ascending powers, where terms are ordered from lowest degree to highest.

Examples

• The expression $p(x) = 7x^4 - 3x^2 + 5$ is a polynomial. Its degree is 4 and its lead coefficient is 7.
• The polynomial $q(x) = 5x - 2x^3 + 8$ written in descending powers is $q(x) = -2x^3 + 5x + 8$.
• The polynomial $r(x) = 4x^3 + x^5 - 9$ written in ascending powers is $r(x) = -9 + 4x^3 + x^5$.

Explanation

A polynomial is an expression built from variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents. The lead coefficient is simply the number in front of the term with the biggest exponent.

Property

The end behavior of a polynomial function determines how the graph extends toward positive and negative infinity. For any polynomial $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$:

• If $n$ is odd and $a_n > 0$: left end goes down, right end goes up
• If $n$ is odd and $a_n < 0$: left end goes up, right end goes down
• If $n$ is even and $a_n > 0$: both ends go up
• If $n$ is even and $a_n < 0$: both ends go down

Examples

Property

The basic cubic function is given by the equation $f(x) = x^3$. Its graph has a characteristic S-shape that passes through the origin. Unlike the parabola $y=x^2$, the cubic function produces negative output values for negative input values.

Examples

• For the function $f(x) = x^3$, the point where $x=-2$ is found by calculating $f(-2) = (-2)^3 = -8$. This gives the guide point $(-2, -8)$.
• The y-intercept of $f(x) = x^3$ is at $x=0$. We have $f(0) = 0^3 = 0$. The graph passes through the origin $(0, 0)$.
• For $f(x) = x^3$, the point where $x=2$ is $f(2) = 2^3 = 8$. This gives the guide point $(2, 8)$, showing how steeply the graph rises.

Explanation

This function cubes its input value. Its S-shaped graph shows that for negative $x$, the output is negative, and for positive $x$, the output is positive. The graph grows faster than a parabola for $x > 1$.