Linear and Non-Linear – Concept & Explanation

Key Concepts

■ Sketch the graph of a linear function

■ Analyze the sketch of a non-linear function

■ Sketch the graph of a non-linear function

Sketch Functions from Verbal Descriptions

Introduction:

In this chapter, we will learn to sketch the graph of a linear function, analyze the sketch of a non-linear function and sketch the graph of a non-linear function 

In the earlier chapter, we have learned to interpret a qualitative graph, interpret the graph of a non-linear function, and describe the relationship of quantities. 

1. What is a qualitative graph? Can you represent the distance traveled over time with a qualitative graph for the below function? 

“Mike is a bicyclist riding along a flat road. He then goes downhill.” 

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

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

Answers: 

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

Qualitative graph with the input variable, the output variable and the intervals: 

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

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

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

3. Graph with increasing, decreasing, or constant functions. 

3.6 Sketch Functions from Verbal Descriptions 

3.6.1 Sketch the Graph of a Linear Function 

Example1: 

Judy is driving a vehicle at a constant speed of 80 km/hr. Draw a distance-time graph to represent the same and then use the graph to find the distance covered by the vehicle in 4 hours 30 minutes. 

Solution: 

Step 1: Identify the two variables. 

Input variable is Q (time). 

Output variable is P (distance covered). 

 Step 2: Analyze the relationship between the two variables. 

The distance is covered at a constant rate over time. 

Make a table. 

Step 3:  

Time (in hours)	1	2	3	4	5
Distance Covered (in km)	80	160	240	320	400

Make ordered pairs from the table. 

 (1, 80), (2, 160), (3, 240), (4, 320), and (5, 400) 

Step 4: Sketch and label a graph that represents the behavior of the function. 

3.6.2 Analyze the sketch of a Non-linear Function 

Example 2: 

Keith tossed a ball in the air and caught it back after few seconds. He sketched the relationship between the height of the ball and time. Describe the behavior of the function in each interval based on his sketch. 

Solution: 

Example 3: 

Jacob is flying a kite in the sky. Describe the behavior of the function in each interval based on his sketch.  

Solution: 

3.6.3 Sketch the Graph of a Non-Linear Function 

Example 4: 

Bryan rides his bike from his home for 40 minutes at a fast pace. He stops to rest for 30 minutes and then continues in the same direction at a slower pace for 40 more minutes. Sketch a graph of the relationship of Bryan’s distance from home over time. 

Solution: 

Step 1: Identify the two variables in the relationship. 

 Input variable: time(t). 

 Output variable: distance from his home(d).   

Step 2: Analyze the relationship between the two variables. 

 When Bryan started his journey, the distance from his home started increasing. 

 The distance remained constant while he was on rest. 

 Then the distance started increasing but at a slow pace.  

 Make a table. 

Time (in minutes)	0	40	70	110
Distance Covered (in km)	0	30	30	50

Step 3: Make ordered pairs from the table. 

 (0, 0), (40, 30), (70, 30), and (110, 50) 

Step 4: Sketch and label a graph that represents the behavior of the function. 

Exercise:

1. Rebecca is driving a two-wheeler continuously at a speed of 20 km/hour. Construct a distance-time graph for this situation.
2. Angela got an offer from an office where she would be paid $25 per hour. What does the graph of a function look like?
3. Matthew works as a salesman in an Audi showroom. He records the number of cars sold in five days (Monday to Friday) on a line graph.

1. Find the total number of cars that were sold in 5 days?
2. On which day were the maximum number of cars sold?
3. How many cars were sold on Wednesday?
4. Which day had the minimum sales of cars?
5. How many more cars were sold on Tuesday than on Monday?

• A biologist conducted a survey on the increasing population of the hippopotamus and recorded his observations on a graph. Answer the following questions.

1) In 2010, find the population of hippopotamuses.

2) Find the total population hippos in 2007.

3) In which year population of hippos increased to 30?

4) Find the minimum recorded year of hippo population.

5) Find the number of hippos recorded in 2008 than in 2006?

• Laura started a local store, and the graph shows the daily earnings of Laura for five days.

• Describe the behavior of the function in each interval based on her sketch.

• The following line graph shows the total number of lions in a zoo.

• Describe the behavior of the function in each interval based on the sketch.

• Diane lives in Baltimore. She measured the temperature every hour for the month of April. Here are her results,

• Describe the behavior of the function in each interval based on the sketch.

• Describe the behavior of the function in each interval based on the sketch.

• What relation between money (in dollars) and time (in months) does this graph show? Write a description of the given graph.

• When a new laptop became available in the store, the number sold in the first week was high. Sales decreased over the next two weeks, and then they remained steady over the next two weeks. The following week, the total number sold by the store increased slightly. Sketch the graph that represents this function over the six weeks.

What have we learned:

■ Sketching the graph of a linear function by identifying the input variable, the output variable and by analyzing the relationship.

■ Analyzing the sketch of a nonlinear function.

■ Sketching the graph of a nonlinear function by identifying the input variable, the output variable and by analyzing the relationship.

Concept Map: