Formative: Average Rate of Change / Functions and their Graphs

Front

What is the average rate of change of a function over an interval?

Back

The average rate of change of a function over an interval [a, b] is calculated as (f(b) - f(a)) / (b - a), representing the change in the function's value divided by the change in the input.

Front

What does it mean for a graph to have a maximum point?

Back

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

Front

How can you determine the interval where a function is 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). This means the graph rises as you move from left to right.

Front

What is the significance of the slope in the context of a graph?

Back

The slope of a graph represents the rate of change of the function. A positive slope indicates an increasing function, while a negative slope indicates a decreasing function.

Front

How do you find the rate of change from a graph?

Back

To find the rate of change from a graph, select two points on the graph, (x1, y1) and (x2, y2), and use the formula (y2 - y1) / (x2 - x1). This gives the average rate of change between those two points.

Front

What does it mean for a population to decrease at the greatest rate?

Back

A population decreases at the greatest rate when the change in population size is the largest negative value over a specific time interval, indicating a rapid decline.

Front

What is the difference between average speed and instantaneous speed?

Back

Average speed is the total distance traveled divided by the total time taken, while instantaneous speed is the speed of an object at a specific moment in time.

Front

What is a function in mathematical terms?

Back

A function is a relation that assigns exactly one output value for each input value. It can be represented as f(x), where x is the input.

Front

What does it mean for a function to be linear?

Back

A linear function is a function that graphs to a straight line, characterized by a constant rate of change and can be expressed in the form y = mx + b, where m is the slope and b is the y-intercept.

Front

How can you identify a maximum or minimum point on a graph?

Back

A maximum point can be identified where the graph changes from increasing to decreasing, while a minimum point is where the graph changes from decreasing to increasing.