Polynomial Graphs

Front

What is the degree of a polynomial function?

Back

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

Front

How do you determine the end behavior of a polynomial function?

Back

The end behavior of a polynomial function is determined by its leading term. If the leading coefficient is positive and the degree is even, the ends of the graph rise. If the leading coefficient is negative and the degree is even, the ends fall. If the degree is odd, one end rises and the other falls.

Front

What is an extremum in the context of polynomial graphs?

Back

An extremum is a point on the graph of a polynomial function where the function reaches a local maximum or minimum value.

Front

What is the maximum number of extrema a polynomial function can have?

Back

The maximum number of extrema a polynomial function can have is one less than its degree. For example, a polynomial of degree 4 can have a maximum of 3 extrema.

Front

What does it mean for a polynomial to have an even degree?

Back

A polynomial has an even degree if the highest power of the variable is an even number. Even degree polynomials have the same end behavior on both sides of the graph.

Front

What does it mean for a polynomial to have an odd degree?

Back

A polynomial has an odd degree if the highest power of the variable is an odd number. Odd degree polynomials have opposite end behaviors on both sides of the graph.

Front

How can you identify the leading coefficient of a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in the polynomial. For example, in f(x) = 3x^5 - 2x^3 + 4, the leading coefficient is 3.

Front

What is the significance of the leading term in a polynomial?

Back

The leading term of a polynomial determines the end behavior and the degree of the polynomial. It is the term with the highest power of the variable.

Front

What is a polynomial function?

Back

A polynomial function is a function that can be expressed in the form f(x) = a_nx^n + a_(n-1)x^(n-1) + ... + a_1x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

Front

What is the difference between a polynomial and a non-polynomial function?

Back

A polynomial function consists of terms with non-negative integer exponents, while non-polynomial functions may include negative exponents, fractional exponents, or variables in the denominator.