Linear Graph

The graphical representation of a linear equation is called a linear graph. The linear graph follows a general equation which is given below:
                

y = mx + c

Where, 

y = dependent variable

x = independent variable

m = slope (rate of change)

c = y-intercept (where the line crosses the y-axis)

The linear graph is represented as a straight line which is shown below:
 

The above graph shows the equation, y = -x + 5.

The slope determines the line’s steepness and direction. A positive slope means an upward trend, and a negative slope means a downward trend. We use linear graphs in science, mathematics, physics and economics.

There are many differences between linear and line graphs, some of them are mentioned below:

Follow the steps given below to plot a linear equation on a graph:

Step 1: Determine the slope and the y-intercept
Example equation: y =2x + 3
m = 2
Y-intercept = 3

Step 2: Plot the y-intercept:
For y = 2x + 3, plot the point at (0, 3)

Step 3: Use the slope to find another point:
Slope = rise/run
For the equation given:
Rise = 2
Run = 1
From (0, 3), move up 2 units and right 1 unit, hence the new point is (1, 5)
Repeat this process to plot additional points for accuracy

Step 4: Draw the line:
Find a ruler and use it to draw a straight line connecting all the points.
Extend the line on both the sides and add arrowheads.

Step 5: Label the Graph:
Label the x-axis and y-axis, and write the equation near the line for clarity.

There are many properties of linear graphs. Some main properties of linear graphs are listed below:

Straight Line:

A linear graph is always a straight line on a coordinated plane

Constant Slope:

The rate of change in a linear graph between any two points always remains the same.

Equation Form:

The linear graph follows the general equation y = mx + c, where:
m is the slope (gradient)
c is the y-intercept.

Intercepts: 

The y-intercept c is the point where the line crosses the y-axis (x = 0). The x-intercept is where the line crosses the x-axis (y = 0).

Linear graphs are very useful and essential for students for various reason which are mentioned below:

Fundamental Understanding: Linear graphs is used to understand the relationships between real-world applications and mathematical variables.

Visualization: They are useful for visualizing equations as it becomes easy for us to interpret changes in data through visualization.

Key Concepts: Linear graphs helps us to understand other important concepts like rate of change, intercepts, and slope.

Problem-Solving Skills: Mastering linear graphs improves the ability of students to solve problems.

Preparation for Advanced Topics: Learning linear graphs provides a strong foundation to learn subjects like algebra, calculus, and statistics.

Students can get confused when solving for linear equations. Here are some tips and tricks to avoid such confusion and learn linear equations easily.

Understanding the Equation Format:

Students must learn and by-heart the standard form of linear equations:

y = mx + c, where m is the slope and c is the y-axis intercept. They must also start noticing how the changes in ‘m’ and ‘c’ affect the graph’s direction and position.

Memorize Key Slope Concepts:

Students must learn the following slope concepts:

Positive slope (m > 0) which means the line rises from the left to the right

Negative slope (m < 0) which means the line falls from the left to the right

A zero slope (m = 0) means the line is horizontal

An undefined slope occurs when the line is vertical, meaning x remains constant, such as x = a.

Plot Key Points:

Students must identify and plot the y-intercept c first. Then they must use the slope (m = rise/run) to find additional points, and then they must connect the points with a straight line.

We use the concept of linear graphs in various applications and fields. Let us now see the different types of uses of the linear graphs:

Business and Economics:

Linear graphs help track revenue and expenses. They also illustrate how price changes affect supply and demand.

Science and Engineering:

In physics, linear graphs represent motion at constant speed (distance vs. time). They also illustrate relationships like Ohm's Law and temperature conversions (Celsius to Fahrenheit).

Medicine and Health:

We use linear graphs to determine the correct medication dosage, it also helps in workouts where a linear increase in heart rate can be plotted during workouts.