End Behavior and Increasing/Decreasing

Front

What is end behavior in functions?

Back

End behavior describes the behavior of a function as the input values (x) approach positive or negative infinity. It helps to understand how the function behaves at the extremes.

Front

Define a turning point of a function.

Back

A turning point is a point on the graph of a function where the function changes from increasing to decreasing or vice versa. It is where the derivative is zero or undefined.

Front

How many turning points can a polynomial function have?

Back

A polynomial function of degree n can have at most n-1 turning points.

Front

What does it mean for a function to be increasing?

Back

A function is increasing on an interval if, for any two points x1 and x2 in that interval, if x1 < x2, then f(x1) < f(x2).

Front

What does it mean for a function to be decreasing?

Back

A function is decreasing on an interval if, for any two points x1 and x2 in that interval, if x1 < x2, then f(x1) > f(x2).

Front

What is the significance of the absolute minimum of a function?

Back

The absolute minimum of a function is the lowest point over the entire domain of the function. It represents the smallest value that the function can take.

Front

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

Back

The end behavior of a polynomial function can be determined by the leading term. If the leading coefficient is positive and the degree is even, both ends go up. If the leading coefficient is negative and the degree is even, both ends go down.

Front

What is the difference between local and absolute extrema?

Back

Local extrema are the highest or lowest points in a specific interval, while absolute extrema are the highest or lowest points over the entire domain of the function.

Front

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

Back

The degree of a polynomial determines the number of turning points and the end behavior. Even degrees have the same end behavior, while odd degrees have opposite end behaviors.

Front

How can you find the turning points of a function?

Back

To find the turning points of a function, take the derivative of the function, set it equal to zero, and solve for x. The solutions are the x-coordinates of the turning points.