Graphing Polynomial Functions

Polynomial graphs are always smooth curves without any breaks, jumps, or sharp corners. When looking at these functions, two key properties determine their overall shape.

The end behavior of a polynomial depends on two factors. First, the degree (highest exponent) tells us if the left and right ends behave the same way or differently. With an even degree, both ends point in the same direction; with an odd degree, they point in opposite directions.

Second, the leading coefficient $the number in front of the highest-powered term$ determines whether the right end points upward or downward. A positive leading coefficient makes the right end go up toward positive infinity, while a negative coefficient sends it down toward negative infinity.

Remember this! A polynomial of degree n can have at most n-1 peaks and valleys (local extrema) and at most n zeros $x-intercepts$.

Zeros of Polynomial Functions

The zeros of a polynomial function are the x-values that make the function equal zero—these appear as x-intercepts on the graph. Finding zeros is a critical skill as they reveal the function's structure.

For a polynomial with degree n, you can have up to n zeros and up to n-1 extrema (high and low points). For example, a third-degree polynomial can have up to 3 zeros and 2 extrema, while a fourth-degree polynomial can have up to 4 zeros and 3 extrema.

The multiplicity of a zero tells us how many times a factor appears in the polynomial. If $x-c$^m is a factor, then c is a zero with multiplicity m. This multiplicity affects how the graph behaves at that x-intercept:

• Odd multiplicity zeros $like x=2 appearing once or three times$ cause the graph to cross through the x-axis
• Even multiplicity zeros $like x=5 appearing twice or four times$ cause the graph to touch but bounce off the x-axis without crossing

Pro tip: When sketching polynomial graphs, identify the zeros first, determine their multiplicity, then add the end behavior to get a surprisingly accurate sketch!

Working with Polynomial Functions

When a polynomial is written in factored form like f(x) = $x+1$$x-3$²$x+5$³, you can immediately identify all the zeros and their multiplicity. In this example, x = -1 has multiplicity 1 (crosses straight through), x = 3 has multiplicity 2 (bounces), and x = -5 has multiplicity 3 (flattens then crosses through).

You can describe a polynomial's complete behavior by noting:

• The degree (sum of all exponents)
• All zeros and their multiplicities
• Whether the graph crosses or bounces at each zero
• The end behavior (where both ends point)

Polynomials can also be created by transforming simple power functions. Starting with basic functions like y = x³ or y = x⁴, you can:

• Shift horizontally by replacing x with $x-h$ or $x+h$
• Stretch or compress vertically by multiplying by a constant
• Reflect over the x-axis by multiplying by -1
• Shift vertically by adding or subtracting a constant

Shortcut: When you see a polynomial in the form a$x-h$ⁿ + k, think of it as the parent function xⁿ that's been shifted, stretched, and possibly reflected!