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 increasing function.

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

Define decreasing function.

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 does it mean if a function has a relative minimum?

Back

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

Front

What does it mean if a function has a relative maximum?

Back

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

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 significance of the leading coefficient in polynomial functions?

Back

The leading coefficient determines the direction of the end behavior of the polynomial function. A positive leading coefficient indicates that the function rises to the right, while a negative leading coefficient indicates that it falls to the right.

Front

What is the difference between absolute minimum and relative minimum?

Back

An absolute minimum is the lowest point over the entire domain of the function, while a relative minimum is the lowest point within a specific interval.

Front

How can you find where a function is increasing or decreasing?

Back

To find where a function is increasing or decreasing, calculate the derivative of the function. Analyze the sign of the derivative: if it's positive, the function is increasing; if negative, the function is decreasing.

Front

What is the role of critical points in determining increasing/decreasing intervals?

Back

Critical points are where the derivative is zero or undefined. They help identify intervals where the function changes from increasing to decreasing or vice versa.