Determining the Equation of a Line

The equation of a straight line can be identified by two possible ways:

1. When a gradient and one point lying on it is given
2. When two points lying on it are given

Gradient is another name for the slope of the line (Maths is fun, 2016).

Calculating Gradient (m)

The gradient (m) of a line can be calculated using the formula:

$$
Gradient (m) =


Change in y-axis


Change in x-axis

For example, if we have two points (1, 2) and (4, 8), we can calculate the gradient as follows:

$$
Change in y-axis (Δy) = 8 - 2 = 6
Change in x-axis (Δx) = 4 - 1 = 3

Now, plug these values into the gradient formula:

$$
Gradient (m) =


6


3


= 2

So, the gradient (m) is 2.

Finding the Intercept (c)

The intercept (c) is the point where the line crosses the y-axis. In the equation of a line, it is represented as:

$$
y = mx + c

For example, if we know the gradient (m) is 4 and a point (2, 6) lies on the line, we can find the intercept (c) as follows:

$$
y = mx + c

6 = 4 * 2 + c
6 = 8 + c
c = 6 - 8
c = -2
So, the equation of the line is:
y = 4x - 2

When two points (2, 8) and (4, 4) are given, we can calculate the gradient (m) as shown earlier, which is -2.

Then, using one of the points, for example (4, 4), we can find the intercept (c) as follows:

$$
y = mx + c

4 = -2 * 4 + c
4 = -8 + c
c = 4 + 8
c = 12
So, the equation of the line is:
y = -2x + 12

Charts and Diagrams

Bar Chart

Bar charts are used to represent data in a graphical format. There are three formats of bar charts:

1. Multiple Bar Chart
2. Simple Bar Chart
3. Component Bar Chart

Multiple Bar Chart

The chart below represents the sales of different juice flavors (Apple, Mango, Mix Fruit) over the years (2017, 2018, 2019):

Juice Flavor	2017 (millions)	2018 (millions)	2019 (millions)
Mango	28	26	27
Mix Fruit	32	34	35
Apple	33	31	30

Simple Bar Chart

The chart below represents the sales of each flavor (Apple, Mango, Mix Fruit) during the month of January:

Juice Flavor	January Month Sales (millions)
Apple	2.25
Mango	2.85
Mix Fruit	2.7

Component Bar Chart

The chart below represents the sales of each flavor (Apple, Mango, Mix Fruit) during every quarter of the year 2018:

Quarter	1st Quarter (Millions)	2nd Quarter (Millions)	3rd Quarter (Millions)	4th Quarter (Millions)
Mango	7.5	7	6.5	7.5
Mix Fruit	8	7	8.5	7.5
Apple	6	8	7.5	6.5

Pie Chart

A pie chart is a circular chart that represents data in wedge-like sectors. Each wedge represents a proportionate part of the whole, and the total value of the pie is always 100 percent.

Below is an example of a pie chart that shows the budget allocated to each flavor of the drink:

Juice Flavor	Budget Allocation
Mango	146,500
Mix Fruit	139,000
Apple	115,500
Total	401,000

Steps to Create the Pie Chart:

1. Convert the budget figures into percentages:

Juice Flavor	Budget Allocation	Percentage
Mango	146,500	36.53%
Mix Fruit	139,000	34.67%
Apple	115,500	28.8%

2. Convert the percentages to degrees (360° total):

Juice Flavor	Percentage	Degrees
Mango	36.53%	131.51°
Mix Fruit	34.67%	124.8°
Apple	28.8%	103.68°

3. Draw the Pie Chart using the calculated degrees.

Line Graph

The following line graph presents the predicted sales during the months of January to June:

Month	Sales forecast (Million)
January	1
February	1.6
March	2
April	2.2
May	1.8
June	1.7

Scatter Graph

Scatter diagrams are used to represent the relationship between two variables. Below is a scatter graph showing the relationship between sugar price and juice price:

Sugar Price (£)	Mango Juice Price (£)
14.1	2.2
14.3	2.4
14.6	2.6
14.7	2.75
15	3

From the scatter graph, we can observe a positive relationship between sugar price and juice price. This suggests that if the sugar prices increase, it will lead to an increase in the price of juice.

Types of Data

Data can be classified into various types:

Qualitative Data

Qualitative data cannot be measured easily; they are often characteristics or descriptors.

Quantitative Data

Quantitative data can be easily measured by numbers.

Binominal Data

Binomial data can be categorized into two mutually exclusive categories, such as true or false, accept or reject, right or wrong.

Ordinal Data

Ordinal data follows a natural order and can be ranked.

Nominal Data

Nominal data is named or labeled and can be classified into various non-overlapping groups.

Discrete Data

Discrete data consists of a finite number of values.

Continuous Data

Continuous data can have an infinite number of values within a range.

Data Examples

Here are some examples of the types of data with their respective categories:

Qualitative Data

Customer feedback (e.g., good or bad)

Quantitative Data

Sales volume (e.g., number of products sold)

Binomial Data

Accept or Reject (e.g., in quality control)

Ordinal Data

Service ratings on a scale of 1 to 5 (e.g., 1 for poor service, 5 for excellent service)

Nominal Data

Preference for pasta types (e.g., Carbonara or Bolognese)

Discrete Data

Number of cars sold (e.g., whole numbers)

Continuous Data

Height of children (e.g., with decimal values like 152.2, 152.3, etc.)

Data Tables

Here are tables summarizing the types of data for various examples:

Qualitative Data

Example	Data Type
Customer Feedback	Qualitative

Quantitative Data

Example	Data Type
Sales Volume	Quantitative

Binomial Data

Example	Data Type
Accept or Reject (Quality Control)	Binomial

Ordinal Data

Example	Data Type
Service Ratings (1 to 5)	Ordinal

Nominal Data

Example	Data Type
Preference for Pasta Types	Nominal

Discrete Data

Example	Data Type
Number of Cars Sold	Discrete

Continuous Data

Example	Data Type
Height of Children	Continuous

These tables provide a clear categorization of the types of data for each example.

Conclusion

In this mathematical paper, we explored the process of determining the equation of a straight line using two different methods: when the gradient and one point are given, and when two points are given. We also discussed the calculation of the gradient (slope) and the intercept of the line, emphasizing the use of the formula "y = mx + c."

Additionally, we delved into various types of charts and diagrams, including bar charts (multiple, simple, and component), pie charts, line graphs, and scatter graphs, providing real-world examples related to a fruit juice company's sales and budget allocation.

Furthermore, we categorized data into different types, such as qualitative, quantitative, binomial, ordinal, nominal, discrete, and continuous, and provided examples for each category.

This comprehensive exploration of mathematical concepts and data representation serves as a valuable resource for understanding and applying these principles in various mathematical and analytical contexts.