Intervals of Increase and Decrease : Functions

Key Concepts

• Interpret a qualitative graph
• Interpret the graph of a non-linear function
• Describe the relationship of quantities

Introduction

• In this chapter, we will learn to interpret a qualitative graph, interpret the graph of a non-linear function  

• Describe the relationship of quantities 

In the earlier chapter, we have learned to compare two linear functions, compare linear function with a non-linear function, and compare properties of two linear functions. 

1. What is meant by the slope of a function? 

2. Identify the slope and write a function from the following graph. 

3. Write the function that represents the below graph 

Answers: 

1. The slope of the function is the change in distance (y) divided by the change in time (x)  

2. The slope is  1.511.51 = 1.5 and the equation is y = 1.5x 

3. The function y = 10(x) + 1 represents the graph. 

Intervals of Increase and Decrease: 

What is a qualitative graph? 

Qualitative graphs represent the relationship between quantities. They are used to represent situations that do not necessarily have any numerical values. 

Qualitative graphs represent the essential elements of a situation in a graphical form.  

It helps to identify the general pattern of behavior of a given function like:  

• Is it linear or not?  
• Do x and y increase together, or does one increase as the other decreases?  
• Is the function increasing or decreasing at a constant rate or not? 

Interpret a Qualitative Graph 

Example1: 

Mike, who is a bicyclist, is riding along a flat road. He then goes downhill. Can you represent the distance traveled over time with a qualitative graph? 

Solution: 

Step 1: Draw a qualitative graph and identify the input variable, the output variable and the  
              intervals. 

Step 2: Determine the relationship between the two variables during the second interval. As time  
              increases, the distance Mike has traveled increases. 

Interpret the Graph of a Non-linear Function 

Example2: 

The graph below represents the temperature inside an oven over a period of time. Describe how the temperature inside the oven and the time are related at each interval. Determine whether the function is increasing, decreasing or constant at each interval. 

Solution: 

The function is increasing at interval 1 because as time increases, so does the temperature inside the oven. 

The function is constant at interval 2 because though the time is increasing, the temperature inside the oven does not change. 

The function is decreasing at interval 3 because as time increases, the temperature inside the oven decreases. 

Example 3: 

The graph below shows the journey of Jim from his home to office. 

Jim started at home at 9 a.m., drove to work, and worked from 10 a.m. until 6 p.m. On the drive back home, Jim got stuck in slow-moving traffic for 2 hours, then stopped to have dinner, and finally got home at 10:30 p.m. 

Describe how the distance and time are related in each interval. 

Determine whether the function is increasing, decreasing or constant in each interval. 

Solution: 

By observing the intervals of the above graph, we can understand the points below: 

• The function is increasing at interval 1 because as time increases, the distance traveled by Jim from home also increases. 
• The function is constant at interval 2 because though the time increases, the distance travelled by Jim is 0. 
• The function is decreasing at interval 3 because as time increases, the distance between Jim office and home decreases. 
• The function is constant at interval 4 because though the time increases, the distance traveled by Jim is 0 as he stopped for dinner. 
• The function is decreasing at interval 5 because as time increases, the distance between Jim’s office and home decreases. 

3.5.3 Describe the Relationship of Quantities 

Example 4: 

Can you identify at which intervals is the function increasing, decreasing, or constant in the graph below? 

Solution: 

Exercise:

1. Frank shot an arrow into the air. The graph below shows the height and distance the arrow traveled in the air during flight.

Identify the input variable, the output variable and the intervals.

2. Laura has observed that the level of water in her sink is decreasing as it drains. Represent this scenario in a graph. Identify whether the function is increasing, decreasing or constant at each interval.
3. Jacob and Nicholas are flying kites in the sky. Jocob’s kite flew for a few minutes, and then suddenly fell to the ground. Draw a qualitative graph.
4. Look at the given graph and answer the following:

• Name an interval where the function is linear.
• Name an interval where the function is increasing.
• Name an interval where the function is decreasing linearly.
• In what interval, the function increases at its greatest rate?

5. Below are the two graphs that look alike. Note that the first graph shows the distance of a car from home as a function of time. The second graph shows the speed of a different car as a function of time. Describe what someone who observes each car’s movement would see in each case.

6. The graph below shows how the altitude of a flight from Boston to Chicago changes over time. Use the graph to answer the following questions.

Describe an interval of x where the plane’s altitude was decreasing.

Describe an interval of x where the plane’s altitude was increasing.

7. The following graph shows the cookie cooling situation.

Determine whether the function is increasing, decreasing or constant in each interval.

8. Determine whether the function is increasing, decreasing, or constant in the following graphs.

9. Identify the increasing and decreasing intervals in the following graph.

10. Use the graph to complete the statements

1. The function is ____________ at intervals 1, 3, and 6.
2. The function is _____________ at intervals 2 and 5.
3. The function is constant at interval _____________

What have we learned:

• Drawing a qualitative graph for the given problem and identifying the input variable, the output variable and the intervals.
• Determining whether the function is increasing, decreasing, or constant at each interval of a non-linear graph.
• Describing the relationship between two quantities by analyzing the behavior of the function relating the quantities at different intervals on a graph.

Concept Map: