Graphs of Polynomials

Front

What is the end behavior of a polynomial function as x approaches -∞ and +∞?

Back

The end behavior of a polynomial function is determined by its leading term. For even-degree polynomials, if the leading coefficient is positive, as x → -∞, f(x) → +∞ and as x → +∞, f(x) → +∞. If the leading coefficient is negative, as x → -∞, f(x) → -∞ and as x → +∞, f(x) → -∞. For odd-degree polynomials, if the leading coefficient is positive, as x → -∞, f(x) → -∞ and as x → +∞, f(x) → +∞. If negative, as x → -∞, f(x) → +∞ and as x → +∞, f(x) → -∞.

Front

What is a polynomial function?

Back

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 and a_n ≠ 0.

Front

How do you determine the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the expression. For example, in the polynomial f(x) = 3x^4 + 2x^3 - x + 5, the degree is 4.

Front

What does the leading coefficient of a polynomial indicate?

Back

The leading coefficient is the coefficient of the term with the highest degree. It affects the end behavior of the polynomial function. A positive leading coefficient indicates that the graph rises to the right, while a negative leading coefficient indicates that it falls to the right.

Front

What is the significance of the roots of a polynomial?

Back

The roots (or zeros) of a polynomial are the values of x for which f(x) = 0. They represent the x-intercepts of the graph and indicate where the graph crosses or touches the x-axis.

Front

What is the relationship between the degree of a polynomial and the number of roots?

Back

A polynomial of degree n can have at most n real roots. However, it may have fewer real roots, and the remaining roots can be complex.

Front

What is the difference between even and odd degree polynomials?

Back

Even degree polynomials have symmetrical graphs and can have a minimum or maximum point. Odd degree polynomials have graphs that extend in opposite directions and have at least one x-intercept.

Front

How can you identify the number of turning points in a polynomial graph?

Back

The maximum number of turning points in a polynomial function is one less than its degree. For example, a 5th degree polynomial can have up to 4 turning points.

Front

What is a factored form of a polynomial?

Back

The factored form of a polynomial expresses it as a product of its linear factors. For example, f(x) = (x - r1)(x - r2)...(x - rn) where r1, r2, ..., rn are the roots of the polynomial.

Front

What is the role of the x-intercepts in graphing polynomial functions?

Back

The x-intercepts of a polynomial function are the points where the graph crosses or touches the x-axis. They are found by solving f(x) = 0.