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Module 14
Sorry! Lines are Busy
This module is designed to make you clearly understand the concept of functions, one of the most important and useful concepts in mathematics.
This module will discuss the notions of domain and range. You will also learn how to sketch graphs of a function that will help you understand the different properties of a function. Finally, the last section of this module will help you determine the characteristics of the graphs of linear functions using the given linear equation and how linear functions relate to your daily activities.
This module has four lessons:
Lesson 1 Functions Defined from a Table of Ordered Pairs Lesson 2 Properties of a Function
Lesson 3 Slope of a Line
Lesson 4 Equation in Slope-Intercept Form
After going through this module, you are expected to:
state the domain and range of a given function;
construct the graphs of a function described by a table of ordered pairs, and vice versa;
identify the different properties of a function such as the intercepts; slope and the trend (decreasing or increasing) from its graph;
define the slope of a line;
express a linear equation in slope-intercept form; and ,
apply linear functions to solve word problems.
What this module is all about
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This is your guide for the proper use of the module: 1. Read the items in the module carefully.
2. Follow the directions as you read the materials.
3. Answer all the questions that you encounter. As you go through the module, you will find help to answer these questions. Sometimes, the answers are found at the end of the module for immediate feedback.
4. To be successful in undertaking this module, you must be patient and industrious in doing the suggested tasks.
5. Take your time to study and learn. Happy learning!
The following flowchart serves as your quick guide in using this module.
How to learn from this module
Start
Take the Pretest
Check your paper and count your correct answers.
Is your score 80% or above?
Yes Scan the items you missed.
No
Study this module
Take the Posttest
3
Before you use this module, take the following Pretest.
Multiple Choice. Read the problems carefully, then choose the correct answer. Write the letter that corresponds to your answer on a separate sheet of paper.
1. Which of the following is the graph of the set of ordered pairs below?
x -2 -1 0
y -2 -4 -6
A. B.
C. D.
2. The domain of the set of ordered pairs {(2,3), (3,4), (5,9), (4,7), (6,6)} is A. {2, 3, 4, 6} B. (3, 4, 6, 7, 9} C. {2, 3, 4, 5, 6|
X Y
X Y
X Y
X Y
4 D. {2, 3, 4, 5}
3. All of the following statements are linear functions EXCEPT one. Which is it? A. y = 3x + 4 B. 2x = 6 C. y = 9 D. xy = 2 4. What is the sign of the slope of the line passing through (-3, 2) and (0, -4)?
A. negative C. both negative and positive
B. positive D. none of these
5. The x and y - intercepts of the equation 3x – 2y = 6 are _____ . A. x = 0; y = 4 C. x = 2; y = -3
B. x = -2; y = -3 D. x = 0; y = -3 6. Which of the given graphs is decreasing?
A. B.
C. D.
7. The zero of the linear function 4x - 3y = 6 is
A. 2/3 B. 3/2 C. 2 D. -2
X Y
X Y
X Y
5
8. The x - intercept of the given graph below is _____.
A. -3
B. 2 C. 3 D. 6
9. If you were to graph 2x + 4y = 8, what would be the slope?
A. - ½ C. 2
B. ½ D. 4
10. What is the slope of the line passing through the points whose coordinates are listed in the table below:
_________________________ | x | 6 | 4 | 2 | 0 |
| y | -3 | 0 | 3 | 6 |
A. 2 B. -2 C. 2 3
D.
3 2
11. The equation in slope-intercept form of the line having the slope m = 3/5 and passing through the point (0, 5) is:
A. y3x5 B. 5 3 5
x
y C. 5
5 3
x
y D. y x
5 3
12. What set of numbers make up the range of the linear function y = x + 2 if the domain is {0, 1, 2, 4}?
A. {0, 2, 3, 4} B. {2, 3, 4, 6} C. {2, 3, 6} D. {2, 4, 6}
13. For children between ages 6 and 10, their height y (in inches) is frequently a linear function of their age x (in years). The height of a certain child is 122 cm at age 6 and 124.5 cm at age 7. What is the slope of the linear function showing the relationship between height and age of children of ages 6 to 10?
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A. -2.5 B. 2.5 C. 1/2 D. -1/2
14. As thunderstorm approaches, you see lightning as it occurs, but you hear the accompanying sound of the thunder a short time afterward. The distance y, in kilometers, that sound travels in t seconds is given by the equation
y = 0.21 t. How long will it take you to hear the thunder from a storm that is 3 kilometers away?
A. 10 seconds B. 14 seconds C. 18 seconds D. 20 seconds
15. Mario collected 25 kilos of cans to recycle. He plans to collect an additional 4 kilos each week. The linear function describing the amount of cans he plans to collect is given by y = 4x + 25. What is the total amount of cans Mario will have collected after 12 weeks?
A. 53 kg B. 37 kg C. 73 kg D. 62 kg
Lesson 1 Functions Defined from a Table of Ordered Pairs
In the previous lesson, you learned how to plot the coordinates of a point in a coordinate plane. In this lesson, you will learn how to construct graphs of linear functions described by a table of ordered pairs and vice-versa.
With a given set of ordered pairs arranged in a table of values and representing the
domain (x) and the range (y), the graph of a function can be drawn.
The set of the first coordinates of the ordered pairs is called the domain of the function. The domain usually contains the x-coordinates. The set of second coordinates is called the range of the function. It usually contains the y-coordinates. Let us begin with a simple example.
What you will do
7
Example 1. How can you graph the function described by the table of ordered pairs that follows?
domain
range
Solution:
a) On a graphing paper, draw a coordinate plane and plot the set of ordered pairs: {(-4, 0), (-2, 2), (0, 4), (2, 6)}
To graph an ordered pair means to draw a dot at the point on the coordinate plane that corresponds to the ordered pair. This is sometimes called plotting a point.
Thus,
b) Draw a line passing through these points. Now place your pencil at the left of the graph to represent a vertical line. Then move the pencil to the right across the graph as shown below:
x -4 -2 0 2
y 0 2 4 6
X Y
(-4,0) (-2,2)
(0,4) (2,6)
X Y
(-4,0) (-2,2)
(0,4) (2,6)
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Observe if for each value of x, the vertical line passes through no more than one point on the graph. This test is called the vertical line test. Thus, the line represents a function. This function represented by the table of ordered pairs is a linear function. The graph of a linear function is a straight line.
c) To determine if the graph represents a function, one needs to perform the vertical line test.
Example 2. Consider the graph below:
From the graph, complete the table by supplying the missing numbers.
x -2 -1 1 2 0
y -1
By looking at the graph, you can easily find the ordinates of the points from the given abscissas. That is, when x = -1, the line passes exactly at -1.
Hence, its ordinate is 0.
How about when x = 1? What is its corresponding ordinate? __________________ Try to draw a vertical line at x = 1, then determine the ordinate of the point where the vertical line intersects the graph. Did you get 2 as its ordinate? _______________ What about when x = 2? What is its ordinate? _____________________________ Also when x = 0? What is its ordinate? ___________________________________
Vertical Line Test:
If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.
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Example 3. Consider the function shown in the graph below:
1. Represent the function shown in the graph as a. a set of ordered pairs
b. a table
2. Determine the domain and range.
Solution: a) The set of ordered pairs for the function is {(-3, 3), (-1, 2), (1,1), (2, -1), (3, -1), (4, 0)} b) The table of ordered pairs:
x -3 -1 1 2 3 4
y 3 2 1 -1 -1 0
c) Domain : {-3, -1, 1, 2, 3, 4} Range : { -1, 0, 1, 2, 3}
A. Graph the set of ordered pairs in the following tables.
1. x -2 -1 0 2 3
y 5 3 1 -3 -5
Self-check 1
X Y
.
.
.
.
10
2.
B. Use the graph to supply the missing numbers in the table. 1.
2.
C. Use the vertical line test to determine if each graph given below is a function.
1. 2.
x -3 -2 -1 1 2
y 5 3 1 -3 -5
x 0 1 2 3 4
y -6 -5 -4
x -1 0 1 2 3
y 5
X Y
X Y
X Y
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3. 4.
Lesson
2
Properties of a Function
Let us consider the graphs below
a. b.
X
Y
X Y
Exploration
X Y
X Y
12
The x-intercepts and y-intercepts of the graph of an equation are its points of intersection with the x-axis and the y-axis. Oftentimes, the abscissa of the
intersection point of the line and the x-axis is called the x-intercept. The ordinate
of the intersection point of the line and the y-axis is called the y-intercept.
At what point does the line in the first graph intersect the x - axis? ________________ the y - axis?___________________
In the second graph, where does the line intersect the x - axis? _______________ the y - axis? __________________
In the first graph, the graph intersects the x-axis at (4, 0). Thus the x- intercept is 4. Since the graph crosses the y-axis at (0, 3), the y-intercept is 3.
What are the x and y-intercepts of the second graph? _____________________________
How do you determine the x - intercepts and y - intercepts? _____________________
Example 1. Use the intercepts to graph the line 4x - 3y = 12.
Solution: Let x = 0 in 4x - 3y = 12 to find the y-intercept. 4x - 3y = 12 original equation 4(0) – 3y = 12
0 – 3y = 12 -3 -3
y = -4
The line intersects the y - axis at (0,-4).
Let y = 0 in 4x - 3y = 12 to find the x-intercept 4x - 3y = 12 original equation 4x – 3(0) = 12
4x – 0 = 12
13 The line intersects the x - axis at (3,0).
To graph the line 4x - 3y = 12, simply plot the x-intercept (3, 0) and the y-intercept (0, -4) and draw the line. That is,
Example 2. Look at the graphs below.
a. b.
The x-intercepts of the first graph and the second graph are respectively 2 and 1. The x-intercepts are obtained by letting y = 0. The values of the x-intercepts are also called the zeros of the function.
Describe the line as it goes from left to right. If the graph of a function rises from left to right, it is increasing. If the line falls from left to right, it is decreasing. Referring to Fig. a
and b, which of the graphs is decreasing? ______________ increasing? ____________ The zero of the function is the value of x when y = 0.
X Y
X Y
X Y
.
.
(0,-4)
14 Fig 1
Fig 2
How do you recognize the rate of change of a linear function in an equation, table, or graph of the function?
Example 3. Consider the graph below:
The steepness and direction of a line in a coordinate plane is determined by its slope. The
slope of a line is the ratio of the change in y-coordinate, or the rise, to the change in x-coordinate, or the run, between two points on the line. Thus,
Slope = change in y-coordinate = rise . change in x-coordinate run
Referring to Fig. 1, in moving from (0, -2) to (1, 0), there is a change of +1 in the x-coordinate and a change of +2 in the y-x-coordinate, or a run of 1 and a rise of 2. So what is the slope of the line drawn in Fig. 1? _____________________________
Move from (1, 0) to (0, -2) as shown in Fig. 2 below: +2 X
Y
(0,-2) (1,0)
+1
X Y
(0,-2) (1,0)
-2
[image:14.612.226.393.184.329.2] [image:14.612.207.374.559.712.2]15 Find the following:
rise? _______________ run? ________________ slope _______________
Thus, if you start at either point and move to the other point, you get the same slope. The graph in Fig. 1, rises from left to right, so it is increasing. Its slope is positive. What is the slope of an increasing function? ______________________________
Take a look of the second graph in Example 2.
In going from A to B, what is the change in y? ______________________ The change in x? _______________________
So, the slope is -1/+1 = -1
The line falls from left to right, so it is decreasing. Its slope is negative. What is the slope of a decreasing function? ________________________
Example 4. Analyze the given graphs below:
a. b.
X
Y
Fig 4
X Y
Fig 5
X Y
A B
Fig 3 (0,1)
[image:15.612.323.499.575.725.2] [image:15.612.103.291.575.727.2]16
Which of the graphs is decreasing? ______________________ Increasing? ________________________________________ What is the slope of the line described in Fig. 4? ______________ How about the slope of the line illustrated in Fig. 5? ____________
What is the relation of the trend (decreasing and increasing) of the graph with the sign of the slope of the line? __________________________
Graph a is increasing, and graph b is decreasing.
If the graph is decreasing, the slope is negative and if the graph is increasing the slope is positive.
A line with a positive slope slants upward from left to right. A line with a negative slope slants downward from left to right.
What is the sign of the slope of the line on the first graph? __________ the second graph?______
A. Complete the table using the graph shown below.
1. 3.
2. 4.
X Y
X Y
X Y
X Y
17
Graph No. y - intercept
slope positive or
negative
x – intercept
or zero of the function
Trend (increasing or
decreasing 1
2 3 4
B. Choose the line whose slope is -1/4.
1. 2.
3. 4.
C. Marina says that only the coordinates of points A and D can be used to find the slope of the line at the right. Mary Fe says she could use the
coordinates of B and D to find the slope. Who is correct? Explain.
Y
X
Y
X
Y
X
Y
X
Y
X
.
. .
.
A B
C
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Lesson 3. Slope of a Line
You have seen graphs where the line rises from left to right, and graphs where the line falls from left to right. Going back to Example 4, graph a rises from left to right, while graph b falls from left to right. Those two graphs show the inclination of the line or the slope of the line.
Since a line is made up of an infinite number of points, you can use any two points on that line to find the slope of the line. So we can generalize the definition of slope for any two points on the line.
Example 1. Determine the slope of the line that passes through (-1, -4) and (3, 2). Sketch the graph.
Solution: Applying the slope formula, we get
2 1 2 1 x x y y m 2 3 4 6 1 3 4 2 ) 1 ( 3 ) 4 (
2
m
Hence, the slope of the line that passes through the points (-1, -4) and (3, 2) is
2 3
.
The graph is given at the right
Given the coordinates of two points (x1, y1) and (x2, y2) on a line, the slope m can be found as follows:
2 1 2 1 x x y y m
where x1 x2
(-1,-4)
(3,2)
Y
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Example 2. Consider the equation 4x - 3y = 12. Find its slope.
Solution: To find the slope of the line with equation 4x - 3y = 12, we make use of the x and y -intercepts. That is,
let x = 0, then y = -4 let y = 0, then x = 3
So the x and y-intercepts of 4x - 3y = 12 are: (3, 0) and (0, -4) respectively. Applying the formula for finding slope, we get
2 1 2 1 x x y y m 3 4 3 4 3 0 0 4
m slope
Hence, the slope of 4x3y12 is 3 4
.
Example 2. Given 2x4y24, find two points and the slope of the line joining them. Solution: The two points are (2, 5) and (6, 3). These are any two points on the line.
2 1 2 1 x x y y m 2 1 4 2 6 2 3
5
m slope
Example 3. Following examples 1 and 2 above, solve for the slope of the line from these two points: (2, 3) and (-2, -2).
Solution: First write the formula, then substitute the given coordinates.
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A. Find the slope of the line represented by the given ordered pairs. Try to graph the line also.
1. (2, -3), (-2, 1) 2. (-3, 1), (3, 5)
B. Determine the slope of each line.
1. 2.
3. 4.
Self-check 3
Y
X
.
.
Y
X
.
.
Y
X
Y
X
.
.
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Lesson
4 Equation in Slope-Intercept Form
If ordered pairs are used to graph a line, an equation can also be used to graph the same line.
An equation of the form Ax + By = C is a linear equation of the first degree. You can transform the given equation into an equivalent equation with y alone on one side. In other words, you solve for y in terms of x.
When the equation is transformed into y = mx + b, the value of y depends on the value of x. So x is called the independent variable and y is the dependent variable.
The equation y = mx + b is called the slope-intercept form because the slope and y-intercept can be read from the equation.
Example 1. Change the equation 2x + y = 3 into the form y = mx + b.
Solution: Solve for y in terms of x. Thus,
2x + y = 3 original equation
2x + (-2x) + y = -2x + 3 Addition Property of Equality
y = -2x + 3
Example 2. Express the equation 3x + 4y = 12 in the form y = mx + b. Solution: Solve for y in terms of x. That is,
3x + 4y = 12 original equation 3x + (-3x) + 4y = -3x + 12
+4y = -3x + 12 y = -3x + 12
4 y =
4 3
x + 3
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Solution: From y = 3x – 1 to the form Ax + By = C
-3x + y = 1 Transpose the term with x in the left side. -1 (-3x + y = -1) Multiply the equation by -1 to make x positive. 3xy1
Example 4. Find the slope and y-intercept of the line 2x - 3y = 4.
Solution: Solve for y to get the slope-intercept form.
2x - 3y = 4 Original equation
-3y = -2x + 4 Subtract 2x from each side 3 4 2 x
y Divide each side by -3 y =
3 2 x - 3 4
Hence, the slope is
3 2
and the y-intercept is (0, 3 4
)
Linear functions are often used to solve real-life problems. Let's have some examples.
Example 5. The time in seconds that a traffic light remains yellow is given by the linear function 1 05 . 0 x
y where x represents the speed limit.
How long will a light remain yellow if the speed limit is 45 km per hour?
Solution: Replace x with 45 in the linear function y = 0.05x + 1. That is, y = 0.05x + 1 given linear function
y = 0.05(45) + 1 y = 3.25 seconds
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Example 6. With a certain long-distance call company, the price of a 4-minute long-distance call is P170. A 11-minute call with the same company costs P415. The linear function that describes the cost of these long-distance calls is y = 35x - 490, where x is the number of minutes consumed for a long-distance call and y is the total cost of charge for the long-distance call. How much would a
20-minute long-distance telephone call cost?
Solution: Simply substitute x = 20 in the linear function y35x30.
30
35
x
y given linear function
30 ) 20 (
35
y
730
y
Therefore, the total cost for a 20-minute long-distance telephone call is P730.
A. Change the following to the form y = mx + b
1. y + x = -1 4. -3x + y = 5 2. 2x + y = 2 5. -4x – y = 6
3. 3x + 4y = 20
B. Write the following equations to the form Ax + By = C 1. y = x + 5 4. y - 1 = x
2. y = -2x + 3 5. 2x = 4 + y 3. y – x = -2
Self-check 4
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The set of all first coordinates from a set of ordered pairs is called the
domain.
The set of all second coordinates from set of ordered pairs is called the
range.
The abscissa of the point where a nonhorizontal line intersects the x-axis is called the x-intercept.
The ordinate of the point where a nonvertical line intersects the y-axis is called the y-intercept.
The slope of a line is defined as
m = change in y-coordinate = rise change in x-coordinate run
Using the coordinates of two points, the slope of a line is given as
m = y2 - y1 where x1 x2 x2 - x1
If the graph of a function rises from left to right, it is said to be increasing
and the slope is positive.
If the graph of a function falls from left to right, it is said to be decreasing
and its slope is negative.
The zero of the function is the value of x when y = 0.
Vertical Line Test for a Function
If any vertical line passes through no more than one point of the graph of a relation, then the relation is a function.
The slope-intercept form of an equation is y = mx + b
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Multiple Choice: Choose the letter of the correct answer.
1. Which of the following tables is represented by the given graph?
A.
B.
C.
D.
2. What is the slope of the line if the graph is increasing?
A. negative C. both negative and positive B. positive D. none of these
3. What is the zero of the function of the linear equation 3x + y = -3?
A. -1 C. -3
B. 1 D. 3
x 0 -1 -3 -2
y 2 1 -1 0
x 2 1 -1 0
y 0 -1 3 -2
x 0 -1 -3 2
y -2 -1 1 0
x 0 -1 3 -2
y 2 1 -1 0
What to do after (Posttest)
Y
26 4. Which is the y - intercept of the graph?
a. –1 b. 0 c. 1 d. 2
5. The range for the set of ordered pairs {(-3, 2), (-2, 2), (-1, 4), (2, -2), (3, -2)} is A. {-1, -2, -3, 2, 3} B. {-2, 2, 4} C. {2, 4}
D. {-2, 2}
6. The x-intercept of the equation 2y - 3x + 1 = 0 is
A. 3 B. -1/2 C. 1/3 D. 1/2
7. What numbers correspond to m and b in the slope-intercept form of the equation
x + 1 = -y?
A. m = -1 ; b = -1 B. m = -1 ; b = 1 C. m = 1 ; b = -1 D. m = 1 ; b = 1
8. Which of the given graphs below has a negative slope?
A. B.
Y
X
Y
X
Y
27
C. D.
9. The equation in the slope-intercept form of the line having a slope of -1/2 and passing through the point (0, -3) is
A. y = 1/2 x + 3 B. y = -1/2 x + 3 C. y = -1/2 x - 3 D. y = 1/2 x - 3
10. The x-intercept and y-intercept of the graph of the equation y = -5x + 5 are
A. (0, -5) and (5, 0) B. (-5, 0) and (0, 5) C. (5, 0) and (0, 5) D. (1, 0) and (0, 5)
11. Each of the graphs of y = 6x - 1, y = x - 1, y = 1/2x - 1 and y = 3x - 1 has
a positive slope and a y-intercept of -1. Which of the graphs has the steepest slope?
A. y = 6x - 1 B. y = x - 1 C. y = 1/2x - 1 D. y = 3x - 1
12. Where does the graph of the equation y = 4x + 3 cross the y -axis? A. -3 B. 3 C. 4 D. -4
13. Running burns about 13 calories per minute. The equation relating t, the number of minutes that you run, and c, the number of calories that you burn is t = 13c.
If you run for 20 minutes, about how many calories will you burn? A. 260 B. 130 C. 60 D. 169
14. Fidel earns P40 for each lawn he mows. The equation y = 40x shows Fidel's earnings (y) and the number of lawns (x) he mows. What is the rate of change of y with respect to x?
A. 1 B. 0 C. 40 D. 1/4
Y
X
Y
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15. A furniture store charges a fee on all items delivered from the store to the customer. The delivery fee (y) is computed by a linear function y = 0.06x + 6, where x denotes the amount purchased. How much will a customer pay for the delivery fee if she will purchase P7000 worth of furniture?
A. P462 B. P420 C. P460 D. P 426
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Pretest page 3 1. B 2. C 3. D 4. B 5. C 6. B 7. B 8. A 9. A 10. C 11. C 12. B 13. B 14. B 15. C
Lesson 1 Self-Check 1 page 9
A.
1. 2.
x
B. 1.
2.
x 0 1 2 3 4
y -6 -5 -4 -3 -2
x -1 0 1 2 3
y 6 3 0 -3 -6
Y
X
Y
30
Lesson 2 Self-Check 2 page 16 A.
Graph No. y - intercept
slope positive or
negative
x – intercept
or zero of the function
Trend (increasing or
decreasing
1 -1 positive 2 Increasing
2 -5 negative -4 decreasing
3 4 positive -3 Increasing
4 3 negative 2 decreasing
B. 2
Lesson 3 Self-Check 3 page 20
A. 1. m = -1 2. 4/6 or 2/3
B. 1.
3 2
2. 2 3
3. 0 4. 1
Lesson 4 Self-Check page 23
A. 1. y = -x – 1
2. y = -2x + 2 3. y = - ¾ x + 5 4. y = 3x + 5 5. y = -4x – 6
Y
X
Y
31 B. 1. x – y = -5
2. 2x + y = 3 3. x – y = 2 4. x – y = -1 5. 2x – y = 4
Posttest page 25 1. A
2. B 3. A 4. B 5. B 6. C 7. A 8. B 9. C 10. C
11. A 12. B 13. A 14. C 15. D
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BIBLIOGRAPHY
Charles, R. I. & Thompson, A.G. (1996). Secondary mathematics: An integrated approach.
USA: Addison-Wesley.
Dossey, J.A. & Embse, C.B. (1996). Secondary Mathematics: An integrated approach. USA:
Addison-Wesley
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