Characteristics of Graphs

Front

What does it mean for a function to be INCREASING on an interval?

Back

A function is INCREASING on an interval if, for any two points x1 and x2 in that interval, where x1 < x2, the function value at x1 is less than the function value at x2 (f(x1) < f(x2)).

Front

What does it mean for a function to be DECREASING on an interval?

Back

A function is DECREASING on an interval if, for any two points x1 and x2 in that interval, where x1 < x2, the function value at x1 is greater than the function value at x2 (f(x1) > f(x2)).

Front

How do you determine the intervals of increase and decrease from a graph?

Back

To determine intervals of increase and decrease from a graph, observe the slope of the graph: if the graph rises as you move from left to right, it is increasing; if it falls, it is decreasing.

Front

What is a maximum point on a graph?

Back

A maximum point on a graph is the highest point in a given interval, where the function value is greater than the values of the function at nearby points.

Front

What is a minimum point on a graph?

Back

A minimum point on a graph is the lowest point in a given interval, where the function value is less than the values of the function at nearby points.

Front

What is the significance of critical points in graph analysis?

Back

Critical points are where the derivative of a function is zero or undefined. They are significant because they can indicate potential maximums, minimums, or points of inflection.

Front

What is the first derivative test?

Back

The first derivative test is a method used to determine whether a critical point is a local maximum, local minimum, or neither by analyzing the sign of the derivative before and after the critical point.

Front

What is the second derivative test?

Back

The second derivative test is a method used to determine the concavity of a function at a critical point. If the second derivative is positive, the function is concave up (local minimum); if negative, it is concave down (local maximum).

Front

What does it mean for a function to have a point of inflection?

Back

A point of inflection is where the function changes concavity, meaning it changes from concave up to concave down or vice versa.

Front

How can you find the maximum value of a function on a closed interval?

Back

To find the maximum value of a function on a closed interval, evaluate the function at the endpoints and at any critical points within the interval, then compare these values.