Analyzing Graphs of Polynomial Functions

Front

What are the roots of a polynomial function?

Back

The roots (or zeros) of a polynomial function are the values of x for which the function equals zero. They are the points where the graph intersects the x-axis.

Front

How does the leading coefficient affect the graph of a polynomial function?

Back

The leading coefficient determines the direction of the graph's ends: if positive, the graph rises to the right; if negative, it falls to the right.

Front

What is a local maximum in a polynomial function?

Back

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

Front

What is a local minimum in a polynomial function?

Back

A local minimum is a point on the graph where the function value is lower than the values of the function at nearby points.

Front

How do you determine the intervals where a polynomial function is increasing?

Back

A polynomial function is increasing on intervals where the derivative is positive.

Front

How do you determine the intervals where a polynomial function is decreasing?

Back

A polynomial function is decreasing on intervals where the derivative is negative.

Front

What is the significance of the degree of a polynomial function?

Back

The degree of a polynomial function indicates the highest power of x and determines the number of roots and the end behavior of the graph.

Front

What does it mean for a root to have an even multiplicity?

Back

A root with an even multiplicity touches the x-axis at that root but does not cross it, resulting in a local maximum or minimum.

Front

What does it mean for a root to have an odd multiplicity?

Back

A root with an odd multiplicity crosses the x-axis at that root, indicating a change in sign of the function.

Front

How can you find the zeros of a polynomial function?

Back

You can find the zeros by factoring the polynomial, using the quadratic formula, or applying synthetic division.