Graphs of Polynomial Functions

Mapping Rule

For a relation \(y = af(k(x - p)) + q\) there is a mapping rule of \((\frac{1}{k}x + p, ay + q)\)

Function Equations

Equation Name	Equation Form	Usage
Expanded form	\(a_nx^n + a_{n - 1}x^{n - 1} + ... + a_1x + a_0\)	\(a_0\) is the y-value of the y-intercept
Factored form	\((x - a_0)(x - a_1)(x - a_2)...(x - a_n)\)	\(a_0, a_1, a_2\) all the way to \(a_n\) are x-values on the x-intercepts (the zeros)
Vertex form	\((x - h)^n + k\)	The function’s vertex is at the point (h, k)

Roots

Roots are when the function is equal to zero. Real roots are also called zeros or x-intecepts. A polynomial function of degree n will have at most n roots.

Order

The number of times a root appears is its order. In \((x - 2)^3\), 2 is a order 3 root. In \((x - 3)(x - 4)\) both 3 and 4 are order 1 roots.

Vertex and Roots

For quadratics, the midpoint of the roots can be used to determine the x value of the vertex.

Describing the Roots

State the number of distinct real roots and say the order of each root. If there are complex roots and the question doesn’t specify, mention those as well.

Key Features

Use the x-intercepts, y-intercepts, degree (odd or even), and end behaviors to produce an image of the graph.