1.2 What is the Business Cycle?

Over time, all economies go through regular cycles of expansion and contraction. This is called the business cycle. 
Today we’ll look at a graph depicting how Gross Domestic Product (GDP) changes with respect to time.

What do you notice about this graph? What do you wonder?

At each of the points A, B, and C, determine if GDP is increasing or decreasing.

The maximum and minimum values of a business cycle are called peaks and troughs.

What can you say about the rate of change of GDP at these peaks and troughs?

Is GDP increasing and decreasing at a constant rate? Explain.

At Point D, GDP is increasing. Is the growth in GDP speeding up or slowing down? How do you know?

When is GDP changing the fastest? How can you tell this from the graph?

At Point E, is the slope of the curve negative or positive? What does this mean in the context of this problem?

At Point E, are the slopes increasing or decreasing? What does this tell you about how GDP is changing at this part of the business cycle?

Describe what is happening at Point F with as much detail as possible.

What is concavity?

Define concavity in your own words.

A function is concave up if the rate of change is increasing.
A function is concave down if the rate of change is decreasing.
A point where a function changes from concave up to concave down or vice versa is called an inflection point, or point of inflection.

Sketch a graph that meets the following requirements:

f(x) is decreasing and concave down for x<-2
f(x) is decreasing and concave up for -2<x<0
f(x) is increasing and concave up for 0<x<3
f(x) is increasing and concave down for x>3

Values for the rate of change of f at the 5 values given in the table below are -4.25, -3, 1, 4, and 9.75. 
Fill in the table below to match each rate of change with the x-value where it occurs.

What do you notice about this graph? What do you wonder?

At each of the points A, B, and C, determine if GDP is increasing or decreasing.

The maximum and minimum values of a business cycle are called peaks and troughs.

What can you say about the rate of change of GDP at these peaks and troughs?

Is GDP increasing and decreasing at a constant rate? Explain.

At Point D, GDP is increasing. Is the growth in GDP speeding up or slowing down? How do you know?

When is GDP changing the fastest? How can you tell this from the graph?

At Point E, is the slope of the curve negative or positive? What does this mean in the context of this problem?

At Point E, are the slopes increasing or decreasing? What does this tell you about how GDP is changing at this part of the business cycle?

Describe what is happening at Point F with as much detail as possible.

What is concavity?

Define concavity in your own words.

Sketch a graph that meets the following requirements:

f(x) is decreasing and concave down for x<-2
f(x) is decreasing and concave up for -2<x<0
f(x) is increasing and concave up for 0<x<3
f(x) is increasing and concave down for x>3

0 < x < 3

(-∞,-1.5)

x>3.5

(-1.5,1.5)

Is f increasing or decreasing at x = −1? How do you know?

Is the rate of change of f increasing or decreasing at x = −1? How do you know?

Approximately on which interval of the domain is the graph of f concave down?

On which interval of the domain is the graph of f concave up?

Estimate the x-value of the point of inflection.

Describe what this point tells you about how f is changing.

1.2 What is the Business Cycle?

What do you notice about this graph? What do you wonder?

At each of the points A, B, and C, determine if GDP is increasing or decreasing.

The maximum and minimum values of a business cycle are called peaks and troughs.

What can you say about the rate of change of GDP at these peaks and troughs?

Is GDP increasing and decreasing at a constant rate? Explain.

At Point D, GDP is increasing. Is the growth in GDP speeding up or slowing down? How do you know?

When is GDP changing the fastest? How can you tell this from the graph?

At Point E, is the slope of the curve negative or positive? What does this mean in the context of this problem?

At Point E, are the slopes increasing or decreasing? What does this tell you about how GDP is changing at this part of the business cycle?

Describe what is happening at Point F with as much detail as possible.

What is concavity?

Define concavity in your own words.

Sketch a graph that meets the following requirements:f(x) is decreasing and concave down for x<-2f(x) is decreasing and concave up for -2<x<0f(x) is increasing and concave up for 0<x<3f(x) is increasing and concave down for x>3

0 < x < 3

(-∞,-1.5)

x>3.5

(-1.5,1.5)

Is f increasing or decreasing at x = −1? How do you know?

Is the rate of change of f increasing or decreasing at x = −1? How do you know?

Approximately on which interval of the domain is the graph of f concave down?

On which interval of the domain is the graph of f concave up?

Estimate the x-value of the point of inflection.

Describe what this point tells you about how f is changing.

Sketch a graph that meets the following requirements:

f(x) is decreasing and concave down for x<-2
f(x) is decreasing and concave up for -2<x<0
f(x) is increasing and concave up for 0<x<3
f(x) is increasing and concave down for x>3