Linear Graph

• What Is a Linear Graph?
• Components of a Linear Graph
• Equation of a Linear Graph
• Properties of Linear Graphs
• Plotting Linear Graphs with Straight Line Equations
• Understanding Linear Graphs and Line Graphs
• Solved Examples
• Practice Problems
• Frequently Asked Questions

What Is a Linear Graph?

All the two-dimensional graphs represent the relationship between two quantities in the form of a graph. The word linear means forming a straight line. 

So, linear graphs are the graphs that represent the relationship between the two quantities in the form of a straight line. It does not include any dot plots, curves, bar graphs. 

For example, consider the following table presenting the points on the `x` and `y`-axis where the `x`-axis represents the quantity of pizza toppings and the `y`-axis represents the cost of toppings.  

The above data can be represented in the form of a graph. This can be done by plotting the points on the cartesian plane. The points representing the `x` and `y`-axes can be plotted as shown in the image below. 

Components of a Linear Graph

1. `x`-axis: The `x`-axis is the horizontal line in a graph that represents the data about the name, place, and other things that are independent in nature. 
2. `y`-axis: The `y`-axis is a vertical line that represents the frequency or measurements of the data on the `x`-axis. This represents the dependent nature of the data, which is dependent on the `x`-axis data. 
3. Label: The label is the name representing the information about the `x`-axis or `y`-axis data. For example, Pizza toppings count on the `x`-axis and topping costs on the `y`-axis. 
4. Chart Title: The chart title represents the information about the chart in a very crisp manner. The chart title must be very short and informative. It should not include any irrelevant information.

Equation of a Linear Graph

A linear graph follows a straight line. A straight line equation is mathematically given as:

`y=mx+b`

where

`m=` slope or gradient of the straight line which determines the steepness or inclination of the line.

`b= y`-intercept of the graph which determines the point where the line crosses the `y`-axis.

Standard form of a linear equation can be written as

`Ax+By=C`

where, `A, B,C` are constants.

Properties of Linear Graphs

1. Linearity: Linear graphs are always straight lines and follow linear equations.
2. Slope or Gradient: A positive slope indicates an upward-sloping line, while a negative slope indicates a downward-sloping line. 
3. Symmetry: A linear graph appears symmetrical when the value of the `y`-intercept `b` is zero. 
4. Infinite solutions: A linear graph has an infinite number of solutions, as each point on the graph represents the solution, and there are an infinite number of points on a line.

Plotting Linear Graphs with Straight Line Equations

Let us draw a line with an equation as follows:

`y=x+1`

Step `1`: Calculate the value of `y` by inserting the values of `x` in the given equation.

For example, if `x=0` then `y=0+1=1`.

 Step `2`: Represent the values in the table as shown above with the points represented in the form of `(x,y)`.

 Step `3`: Plot the points on the graph.

 Step `4`: Join the points on the graph to form a straight line.

Graph a Line Given the Equation in Slope-Intercept Form

Here are the steps to draw a straight line on a graph using the slope-intercept form.

Step `1`: Identify the slope (\(m\)) and the `y`-intercept (\(b\)).

Step `2`: Plot the y-intercept on the `y`-axis.

Step `3`: Use the slope to find additional points on the line. For example, if the slope is \(2\), move up by \(2\) units vertically and \(1\) unit horizontally from the `y`-intercept to find another point.

Step `4`: Connect the points to draw a straight line.

Example: Graph the linear equation: \( y = 2x + 3 \)

`1`. Identify the slope and `y`-Intercept:

• Slope (\(m\)) `= 2`
• `Y`-Intercept (\(b\)) `= 3`

`2`. Plot the `y`-Intercept:

• Plot the point `(0, 3)` on the `y`-axis.

`3`. Use the Slope to Find Another Point:

• We can write the slope `2` as \(\frac{1}{2} \), indicating \(\frac{\text{rise}}{\text{run}} \). 
• Continue from the `y`-intercept, move up `2` units and to the right `1` unit to find the next point `(1, 5)`.

`4`. Connect the Points:

• Draw a straight line through the two plotted points.

Understanding Linear Graphs and Line Graphs

Linear Graph: A linear graph is a graph that follows the line equation `y=mx+b`. This is plotted on the cartesian coordinate in the form of a straight line to represent the linear relationship between the two axes.

Line Graph: A line graph is a method of data visualization in statistics. it is used to show a relationship between the two axes, i.e., the `x` and `y` axes. It is a collection of line segments that represents a data.

Solved Examples

Example `1`: Consider the linear equation `y=4x+2`. Determine the slope and the y-intercept of the line.

Solution: 

Comparing with the equation of line `y=mx+b`

we can find that slope `(m)=4`

and `y`-intercept `(b)=2`

Example 2: What is the slope of the given linear graph?

Solution:

In the given graph, let's select two points: \( A \) with coordinates \( (2, 4) \) and \( B \) with coordinates \( (6, 10) \).

Now, let's calculate the change in \( y \) and the change in \( x \):

Change in \( y \) = \( 10 - 4 = 6 \)

Change in \( x \) = \( 6 - 2 = 4 \)

Next, we'll apply the slope formula:

\[ \text{slope} = \frac{{\text{change in } y}}{{\text{change in } x}} = \frac{6}{4} = 1.5 \]

Example `3`: Determine the nature of the line represented by the equation \( y = 2 \).

Solution:

The equation \( y = 2 \) represents a horizontal line because the value of \( y \) is always `2`, regardless of the value of \( x \).

Example `4`: Determine whether \( y = -3x + 2 \) is a downward-sloping line or an upward-sloping line.

Solution:

Slope-intercept form: `y=mx+b`

Compare the given equation with the slope-intercept form to find the slope `(m)`.

`m=-3`

The line represented by the equation \( y = -3x + 2 \) is a downward-sloping line because slope is negative.

Practice Problems

Q`1`. Identify the graph of `y=2x+4`.

Answer: a

Q`2`. Find the slope and y-intercept of \( y = -\frac{3}{4}x + 5 \).

1. Slope: \(\frac{3}{4}\), `y`-Intercept: `5`
2. Slope: `-3, y`-Intercept: `4`
3. Slope: \(-\frac{3}{4}\), `y`-Intercept: `5`
4. Slope: \(\frac{4}{3}\), `y`-Intercept: `-5`

Answer: c

Q`3`. Identify the slope of the line represented by the equation \( y = 2x + 4 \).

1. Downward-sloping line
2. Upward-sloping line
3. Vertical line
4. Horizontal line

Answer: b

Q`4`. Determine the nature of the line represented by the equation \( x = 2 \).

1. Vertical line
2. Horizontal line
3. Upward-sloping line
4. Downward-sloping line

Answer: a

Frequently Asked Questions

Q`1`. What is the slope-intercept form of a linear equation?

Answer: The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the `y`-intercept.

Q`2`. Explain the significance of the slope in a linear graph.

Answer: The slope (\(m\)) represents the rate of change or steepness of the line. A positive slope indicates an upward-sloping line, while a negative slope indicates a downward-sloping line. A slope of zero represents a horizontal line and an undefined slope represents a vertical line.

Q`3`. What does the y-intercept represent?

Answer: The y-intercept is the point where the line intersects the y-axis. In the equation \(y = mx + b\), \(b\) is the `y`-intercept. 

Q`4`. Describe the characteristics of a horizontal line and a vertical line on a linear graph.

Answer: A horizontal line has a slope of zero and extends left and right along the `x`-axis. A vertical line has an undefined slope and extends infinitely up and down along the `y`-axis.