Exercise 2.2: Graphs, Intersections, and Applications | First Year Math (New 2025 Edition)

Exercise 2.2: Graphs, Intersections, and Applications | First Year Math (New 2025 Edition)

Welcome to MathUniverseOnline.com, your trusted online source for quality math notes, exam preparation materials, and concept-based explanations. In this in-depth article, we are focusing on Exercise 2.2 from the First Year Mathematics Book (New 2025 Edition), which deals with functions and their graphical representation.

This exercise helps students transition from solving algebraic equations on paper to visualizing relationships through graphs. It includes topics such as intersection points of graphs, how linear functions behave when plotted, and how these mathematical ideas apply in the real world, including finance and science.
Let’s explore each section of this exercise in detail.

                                              

                                                         

Understanding Graphs and Intersections
Before diving into individual topics, it’s important to understand what a graph really is. A graph is a visual representation of a function or an equation. It shows how one variable changes when another one changes.
For example, if you're plotting the relationship between hours studied and marks gained, a graph helps you see that relationship clearly—more study usually means more marks, and this appears as an upward sloping line.
Now let’s study the key parts of Exercise 2.2.

1. Finding Intersecting Points by Drawing Graphs

When two graphs are drawn on the same coordinate plane, the point where they meet is known as the point of intersection. This point represents the value of the variables that satisfy both equations at the same time.
Steps to Find Intersection Graphically:
Choose at least four or five values for x (horizontal axis) for each equation.
Calculate the corresponding values for y (vertical axis).
Plot the points on graph paper.
Draw the curves or straight lines for each function.
Look for the point(s) where the graphs cross each other.

This method provides a clear, visual solution to problems that may be difficult to solve algebraically. It also improves the student's ability to interpret graphs effectively.

2. Intersection of a Linear Function and the Coordinate Axes

A linear function is usually written in the form of a straight line. When we plot it on graph paper, it will always cross the x-axis and the y-axis—unless it's a special case like a vertical or horizontal line.
To find where the line meets the x-axis:
Set y equal to zero and find the value of x.

To find where the line meets the y-axis:
Set x equal to zero and find the value of y.

These two points are called the x-intercept and y-intercept, and they are extremely helpful in drawing the graph of a straight line quickly and accurately.
Knowing where a line cuts the axes helps students understand the starting value and the zero point of many real-life situations such as starting salary, initial investment, or base level in physics.


3. Intersection of Two Linear Functions

When two straight lines are plotted on the same graph, their point of intersection shows the values of x and y where both lines are equal.
There are three possible outcomes:
1. The lines cross at one point – this means there is a unique solution to the system.

2. The lines are parallel – there is no solution.

3. The lines are identical – there are infinitely many solutions.


By plotting both lines and identifying the intersection visually, students gain a strong understanding of systems of equations, which are frequently used in economics, engineering, and physics.


4. Intersection of a Linear Function and a Quadratic Function

A quadratic function is a curve that opens upward or downward. When a straight line and a quadratic curve are drawn together, there can be:
Two points of intersection (line crosses the curve at two places),
One point of intersection (line touches the curve at exactly one point),
Or no point of intersection at all.

This part of the exercise helps students see how non-linear and linear functions interact with each other. It introduces the concept of discriminants and roots, though not explicitly, by showing visually how many solutions are possible.
In real life, such interactions can model problems in physics (motion under gravity), business (profit vs. cost), and engineering (load vs. strength).


5. Graph of the Square Root Function

The square root function behaves differently from linear and quadratic functions. Its graph:
Starts at the origin (zero, zero),
Exists only for zero and positive x-values,
Grows slowly as x increases.

Key Features:
It is not symmetric.
The curve gradually flattens out as x becomes large.
The domain (input values) is zero or greater.
The range (output values) is also zero or greater.

This function is important in science, especially in problems related to energy, sound, and light. It also appears in statistics, where it is used in standard deviation and error analysis.


6. Graph of the Cube Root Function

Unlike the square root function, the cube root function can accept both positive and negative values of x. Its graph:
Passes through the origin,
Extends in both directions (left and right),
Is symmetric with respect to the origin.

Key Features:
Domain: All real numbers.
Range: All real numbers.
The graph looks like a stretched “S” curve.

This type of function appears in geometry, volume measurements, and physics, especially when calculating the behavior of gases or fluids.

7. Real-Life Applications of Graphs

Graphs are not just limited to textbooks—they are essential tools used in many fields.
In Physics:
Velocity and acceleration are understood using graphs.
Motion, distance, and time relationships are shown graphically.

In Economics:
Supply and demand graphs show the equilibrium price.
Cost and revenue graphs help businesses find break-even points.

In Medicine:
Graphs show the rise and fall of heart rates and brain activity.
They are used in lab reports and medical monitoring.

In Technology:
Graphs are used in coding, data science, and software development.
Machine learning algorithms use graphs to predict behavior.

Graphical understanding is key to analyzing patterns, trends, and relationships in a wide variety of professions.

8. Growth and Decay in Finance (Predicting Long-Term Stock Prices)

This is perhaps the most practical application of functions and graphs. Finance relies heavily on the ability to predict how quantities change over time, especially in areas like investment, savings, loans, and stock prices.
Growth Example:
When money is invested, it grows over time at a certain percentage. The graph of such an investment goes upward over time.
For instance, if you invest a certain amount in stocks and it grows yearly, you can use a graph to predict the total amount after five or ten years.
Decay Example:
In some cases, the value of an item goes down, such as the price of a car or machinery. This is called depreciation. The graph of this situation shows a downward curve, getting lower over time.
Stock price predictions also use similar models, where the market is analyzed based on previous data. By using graphs, financial analysts can predict:
Whether the price will rise or fall,
When to invest or withdraw money,
How much profit or loss might occur in future.

Graphs are powerful tools in forecasting, investment analysis, and long-term planning.

PDF SOLUTION :

                                     

                          

Conclusion:

Exercise 2.2 of the First Year Math 2025 book is not just an academic lesson; it is a real-world skill builder. Through this exercise, students learn how to:
Draw graphs of linear, quadratic, square root, and cube root functions,
Understand points of intersection between different types of graphs,
Apply mathematical concepts in physics, business, and finance.

By understanding the visual nature of equations, students become better problem solvers and are able to connect math with real life.
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