Category: Tables, Graphs, Functions

Introduction

Many real-world situations involve two variable quantities in which one quantity depends on the other. The quantity that depends on the other quantity is called the dependent variable, and the quantity it depends on is called the independent variable. The values of variables are used in tables and in plotting graphs.

In this lesson, we identify the dependent variable and the independent variable in a table or a graph.

Example 1

Identify the dependent and independent variables in the following problem.

Solution

Independent variable is Clay bought by teacher; Dependent variable is Clay available for class

Example 2

Identify the dependent and independent variables in the following problem.

Time in years versus the weight in pounds is shown in the table below.

Time in yearsx	0	1	2	3
Weighr in poundsy	0	60	120	180

Solution

Independent variable is Time in years; Dependent variable is Weight in pounds.

Introduction

In this lesson, we solve real world problems which are modeled by an equation. The equation describing the problem is written and its graph is plotted.

Example 1

Jenny has savings of 42andearns7 for each hour of lawn mowing. If y is the total amount with Jenny and x is the number of hours he works, write and equation in y and x and graph it.

Solution

Step 1:

The equation representing the problem in x and y is

y = 42 + 7x

Step 2:

The plot is

Example 2

Phil’s party costs 110plus7 for every guest he invites. Let y be the total cost of party and x be the number of guests. Write an equation in x and y and graph it.

Solution

Step 1:

The equation representing the problem in x and y is y =110 + 7x

Step 2:

The plot is

Introduction

In this lesson, given a table of values, graphs are plotted from ordered pairs and equations are written.

Suppose we are given a table showing x and y values. The independent variable is represented by x and the dependent variable is represented by y and we have the ordered pairs of x,y.

• From the ordered pairs, a graph is plotted
• From the table of values, an equation is found giving the relation between x and y.

Example 1

Plot the ordered pairs described by the table. Write an equation relating x and y.

Inputx	Outputy
0	0
1	12
2	24
3	36
4	48

Solution

Step 1:

The equation of the ordered pairs of given table is

y = 12x

Step 2:

The plot of the ordered pairs of given table −

Example 2

Plot the ordered pairs described by the table. Write an equation relating x and y.

Inputx	Outputy
0	0.5
4	1.5
8	2.5
12	3.5
16	4.5

Solution

Step 1:

The equation of the ordered pairs of given table is

y = 0.5 + x4

Step 2:

The plot of the ordered pairs of given table −

Introduction

In this lesson, given a real-world situation, we write a function to model the problem and then evaluate it for a particular situation or value.

Example 1

Moe has savings of 70andheearns5 for each hour of lawn mowing. If A is the amount with Moe and h is the number of hours he works, write an equation in A and h. Find how much amount he has after 6 hours of mowing lawn.

Solution

Step 1:

Equation in A and h, A = 70 + 5h; h = 6

Step 2:

A = 70 + 5h = 70 + 56 = 70 + 30 = $100;

So, total amount with Moe = A = $100

Example 2

Stacy is putting 270inasavingsaccountandadding40 each week. Let S represent the total amount saved and let w represent the number of weeks Stacy has been adding money. Write an equation relating S and w and use it to find the total amount after 12 weeks.

Solution

Step 1:

Equation in S and w, S = 270 + 40w; w = 12

Step 2:

S = 270 + 40w = 270 + 4012 = 270 + 480 = $750;

So, Total amount = S = $750

Introduction

In this lesson, we have two-step functions with decimals modeling real world problems. In such a case we find outputs of those functions.

Example 1

The amount that Jane has is given by the function Ax = 0.8x + 18, where x is her allowance in dollars. What is the amount she has if her allowance is $15?

Solution

Step 1:

Amount with Jane, Ax = 0.8x + 18;x=15

Step 2:

Ax = 0.8x + 18=0.8(15)+18 = 12.0+18 = $30

So, Ax = $30

Example 2

The sum of three consecutive numbers is given by the function Sn = 3n + 3 where n is the smallest number. If the smallest of the three numbers is 38 what is the sum of the three consecutive numbers?

Solution

Step 1:

Sum of numbers, Sn = 3n + 3; n = 38

Step 2:

Sn = 3n + 3 = 338 + 3 = 114 + 3 = 117

So, S38 = 117

Introduction

In this lesson, we have one-step functions modeling real world problems. In such cases we find outputs of those functions.

Example 1

After eating at a restaurant, Joe, Tom, Bill and Sunny decided to divide the bill evenly. If the total bill given by the function Ax = 4x amounted to $136, how much should each pay(x)?

Solution

Step 1:

Total bill amount Ax = 4x = $136

Step 2:

4×4 = $ 1364

So x = $34

Example 2

Shania is baking a cake. The recipe calls for Cy = y + 3 cups of flour. How many cups does she need if y = 5 cups?

Solution

Step 1:

Cups of flour, Cy = y + 3 cups; y = 5

Step 2:

Cy = y + 3 = 5 + 3 = 8

So C5 = 8 cups

Introduction

A line graph is mostly used to show change over time as a series of data points connected by line segments on the coordinate plane. The line graph therefore helps to find the relationship between two data sets, with one data set always being dependent on the other set.

Line graphs are drawn such that the independent data values are on the x-axis and the dependent data values are on the y-axis. Line graphs are used to track changes over short and long periods of time or some independent variable.

Let’s define the various parts of a line graph.

S.No.	Part & Description
1	TitleThe title of the graph tells us what the graph is all about.
2	LabelsThe horizontal label across the bottom and the vertical label along the side tells us what kinds of data is being shown.
3	ScalesThe horizontal scale across the bottom and the vertical scale along the side tell us how much or how many.
4	PointsThe points or dots on the graph represents the x,y coordinates or ordered pairs.
5	LinesThe line segments connecting the points give estimated values between th points.

Uses of line graphs

Line graphs are useful in that they show data variables and trends clearly and help us make predictions about the results of data not yet included. They can also be used to show several dependent variables against one independent variable. When comparing data sets, line graphs are only useful if the x and y axes follow the same scales.

Interpreting line graphs

We interpret line graphs by studying and analysing data from line graphs. Interpreting the line graph data is

• Making sense of the given data
• Answering queries about the data
• Making predictions on trends,
• Finding value of one variable given the value of the other and so on.

Example 1

To monitor the health of her potato plants, Ms. Fiona recorded the number of potatoes that grow in her garden each year. In which year on the graph did the largest number of potatoes grow in Ms. Fiona’s garden?

Solution

It is found from the graph above that the largest number of potatoes grew in the year 2011.

Example 2

The graph below shows the rainfall over a period of 12 months. Find the month in which there was the second least rainfall.

Solution

From the graph, the month in which there was the second least rainfall was August.

Introduction

We graph a line whose equation is given, say for example y = 3x. We need at least two points or ordered pairs to graph the line. First we choose some x values. Then we evaluate y = 3x for each value of x.

For example for x = 0, y = 30 = 0; x = 1, y = 31 = 3 and so on. We put the x, y and the ordered pair x,y values as follows.

x	y	x,y
0	30 = 0	0,0
1	31 = 3	1,3
2	32 = 6	2,6
3	33 = 9	3,9
4	34 = 12	4,12

We see that the ordered pairs lie in the quadrant 1. Joining the points will give the graph of the line in quadrant 1.

Example 1

Graph the line in the quadrant 1, whose equation is given below.

2x y = 3

Solution

Step 1:

Given equation 2x y = 3; for y = 0, x = 3/2; for x = 2, y = 223 = 43 = 1. So two ordered pairs are 3/2,0 2,1

Step 2:

Plotting the points and joining with a line we get

Example 2

Graph the line in the quadrant 1, whose equation is given below.

x + 3y = 1

Solution

Step 1:

Given equation x + 3y = 1; for y = 0, x = 1; For x = 0, y = 1/3. So two ordered pairs are 1,0 and 0,1/3

Step 2:

Plotting the points and joining with a line we get

Print Page

Introduction

In this lesson, we find the function rule given a table of ordered pairs.

We first identify the input and the output variables and their values. We find if the function is increasing or decreasing.

If the function is increasing, it means there is either an addition or multiplication operation between the two variables.

If the function is decreasing, it means there is either a subtraction or division operation between the two variables.

Consider the following table −

x	y
3	15
5	25
6	30
8	40
9	45

We see that the y values are increasing as the x values are increasing. So it is an increasing function. So, the variables x and y must be related either by addition or multiplication operation.

We check addition operation on x and y values as follows −

3 + 12 = 15

5 + 12 = 17

We check multiplication operation on x and y values as follows −

3 x 5 = 15

5 x 5 = 25 and so on

We see that the relation between x and y is a multiplication operation here and the constant for which all values are satisfied is 5.

So the function rule for this table of x and y values is Multiply by 5.

Consider another table −

x	y
10	13
15	18
19	22
23	26
28	31

Here we identify the input and output and then see the output y is increasing as input x is increasing.

13 = 10 + 3; 18 = 15 + 3; 22 = 19 + 3 and so on.

So, output y = input x + 3

Therefore, we identify the function rule here as Add 3.

Example 1

Given the following table of ordered pairs, write a one-step function rule.

Inputx	Outputy
0	3
2	5
4	7
6	9
8	11

Solution

Step 1:

From the table 0 + 3 = 3; 2 + 3 = 5 and so on

Step 2:

Input + 3 = Output or x + 3 = y

Step 3:

Therefore the function rule here is Add 3 to the input to get the output.

Example 2

Given the following table of ordered pairs, write a one-step function rule.

Inputx	Outputy
0	0
1	6
2	12
3	18
4	24

Solution

Step 1:

From the table 0 × 6 = 0; 1 × 6 = 6 and so on

Step 2:

Input × 6 = Output or x × 6 = y

Step 3:

Therefore the function rule here is Multiply by 6 the input to get the output.

Introduction

In this lesson, we fill in a table with x and y values given a function rule with two-step rules. Usually the function rule is a linear equation in x and y. By giving different values to the independent variable x, corresponding values of dependent variable y are found. These x and y values are then put in a table. Thus a function rule with two-step rules is used to fill in a table with x and y values.

Example 1

Fill in the table using this function rule.

y = 5x 3

Solution

Step 1:

In y = 5x 3, for x = 0, y = 50 3 = 3; for x = 1, y = 51 3 = 2;

Similarly, for x = 2, y = 7; for x = 3, y = 12 and for x = 4, y = 17.

Step 2:

Making a table of x and y values we get

Inputx	Outputy
0	-3
1	2
2	7
3	12
4	17

Example 2

Fill in the table using this function rule.

y = 3x 3

Solution

Step 1:

In y = 3x 3, for x = 0, y = 30 3 = 3; for x = 1, y = 31 3 = 0;

Similarly, for x = 2, y = 3; for x = 3, y = 6 and for x = 4, y = 9.

Step 2:

Making a table of x and y values we get

Inputx	Outputy
0	-3
1	0
2	3
3	6
4	9