Unit 4 - Polynomial Graphs

Front

What is the end behavior of a polynomial graph as x approaches infinity?

Back

The end behavior of a polynomial graph as x approaches infinity depends on the leading coefficient and the degree of the polynomial. If the leading coefficient is positive and the degree is even, then the graph approaches infinity. If the leading coefficient is positive and the degree is odd, the graph approaches infinity on the right and negative infinity on the left.

Front

What are the zeros of a polynomial function?

Back

The zeros of a polynomial function are the values of x for which the polynomial equals zero. They can be found by factoring the polynomial or using the quadratic formula for quadratic functions.

Front

What is a double root in a polynomial function?

Back

A double root is a root that has a multiplicity of two. This means that the factor corresponding to this root appears twice in the polynomial. The graph of the polynomial will touch the x-axis at this point but will not cross it.

Front

How do you find the factors of a polynomial given its zeros?

Back

To find the factors of a polynomial given its zeros, use the fact that if r is a zero, then (x - r) is a factor. For example, if the zeros are -3, 2, and 8, the factors are (x + 3), (x - 2), and (x - 8).

Front

What is a local maximum point in a polynomial graph?

Back

A local maximum point is a point on the graph of a polynomial where the function value is higher than the values of the function at nearby points. It represents a peak in the graph.

Front

What is the degree of a polynomial?

Back

The degree of a polynomial is the highest power of the variable in the polynomial expression. It determines the shape and end behavior of the graph.

Front

What is the leading coefficient of a polynomial?

Back

The leading coefficient is the coefficient of the term with the highest degree in a polynomial. It influences the end behavior of the graph.

Front

How do you determine the number of zeros a polynomial can have?

Back

The Fundamental Theorem of Algebra states that a polynomial of degree n can have at most n real zeros. The actual number of real zeros can be fewer than n.

Front

What is the significance of the x-intercepts of a polynomial graph?

Back

The x-intercepts of a polynomial graph are the points where the graph crosses or touches the x-axis. They correspond to the zeros of the polynomial.

Front

What is the relationship between the degree of a polynomial and its end behavior?

Back

The degree of a polynomial determines its end behavior. Even-degree polynomials have the same end behavior on both sides, while odd-degree polynomials have opposite end behaviors.