Lesson 1: Graphing Polynomial Functions

Property

• The degree of a term is the sum of the exponents of its variables.
• The degree of a constant is $0$.
• The degree of a polynomial is the highest degree of all its terms.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial.

Examples

• The polynomial $2x^5 - 3x^2 + 7$ has a degree of $5$, which is the highest power of $x$.

Property

For polynomial function $f(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0$ where $a_n \neq 0$:

Odd degree ($n$ is odd):

Property

A polynomial function of degree $n$ has at most $n-1$ turning points. A turning point is a point where the graph changes from increasing to decreasing or vice versa. These points correspond to a relative maximum (the highest point in a particular section of the graph) or a relative minimum (the lowest point in a particular section of the graph).

Examples

• The function $f(x) = x^3 - 3x + 1$ has a relative maximum at $(-1, 3)$ and a relative minimum at $(1, -1)$. Since the degree is $3$, it can have at most $3-1 = 2$ turning points.
• The function $g(x) = x^4 - 8x^2 + 2$ has relative minima near $(-2, -14)$ and $(2, -14)$, and a relative maximum at $(0, 2)$. This degree $4$ polynomial has $3$ turning points, which is less than or equal to $4-1=3$.

Explanation

Turning points are crucial features for sketching the graph of a polynomial. A relative maximum occurs where the function''s values change from increasing to decreasing, creating a "peak". Conversely, a relative minimum occurs where the values change from decreasing to increasing, creating a "valley". The total number of these peaks and valleys is limited by the degree of the polynomial, which helps in verifying the shape of your graph.

Property

When polynomial functions model real-world situations, the domain must be restricted to values that make sense in context. Always consider physical constraints, time limitations, and meaningful ranges when interpreting polynomial models.

Examples