For each polynomial function, identify its graph from choices A–F. ƒ(x)=(x-2)(x-5)

Key Concepts

Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. The general form is f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where 'n' is a non-negative integer. Understanding polynomial functions is crucial for analyzing their graphs, as they exhibit specific characteristics based on their degree and leading coefficient.

Factored Form

The factored form of a polynomial expresses it as a product of its linear factors. For example, the function f(x) = (x-2)(x-5) indicates that the roots of the polynomial are x = 2 and x = 5. This form is useful for identifying the x-intercepts of the graph, which are the points where the graph crosses the x-axis.

Graphing Polynomial Functions

Graphing polynomial functions involves plotting points based on the function's values and understanding its behavior at critical points, such as intercepts and turning points. The degree of the polynomial determines the number of turns in the graph, while the sign of the leading coefficient influences the end behavior. For the given function, the graph will intersect the x-axis at its roots and will be a smooth curve.