Linear Graphs

 

Linear Graphs

A linear graph is a visual representation of a linear equation or a relationship between two variables. It is a straight line that shows how one variable changes in relation to the other, typically in the form of the equation:

$$y = mx + b$$

where:

• $y$ is the dependent variable (usually plotted on the vertical axis),
• $x$ is the independent variable (usually plotted on the horizontal axis),
• $m$ is the slope of the line, which represents how much $y$ changes for a given change in $x$,
• $b$ is the y-intercept, which is the value of $y$ when $x = 0$.

Key Elements of a Linear Graph:

1. Slope ($m$):

   • The slope determines the steepness of the line. It is calculated as the ratio of the change in $y$ to the change in $x$, or: $$m = \frac{{\Delta y}}{{\Delta x}}$$ A positive slope means the line rises from left to right, and a negative slope means the line falls from left to right.

2. Y-Intercept ($b$):

   • This is the point where the line crosses the y-axis. If $x = 0$, then the value of $y = b$.

3. Coordinates:

   • Any point on the graph is represented by a pair of values $(x, y)$.

How to Plot a Linear Graph:

1. Start with the y-intercept ($b$):

   • Mark the point where the line crosses the y-axis. This point will be $(0, b)$.

2. Use the slope ($m$):

   • From the y-intercept, use the slope to determine another point on the line. The slope tells you how much $y$ changes when $x$ changes. For example, if $m = 2$, this means that for every 1 unit increase in $x$, $y$ will increase by 2 units.

3. Draw the Line:

   • Once you have two points (at least), draw a straight line through them, and extend the line in both directions.

Example 1:

Consider the equation $y = 2x + 1$:

• The slope $m = 2$, so for every 1 unit increase in $x$, $y$ increases by 2 units.
• The y-intercept $b = 1$, so the line crosses the y-axis at the point $(0, 1)$.

Plotting two points:

• At $x = 0$, $y = 1$ (point $(0, 1)$).
• At $x = 1$, $y = 2(1) + 1 = 3$ (point $(1, 3)$).

Now, draw a straight line passing through these points.

Example 2:

Consider the equation $y = -3x + 4$:

• The slope $m = -3$, meaning for every 1 unit increase in $x$, $y$ decreases by 3 units.
• The y-intercept $b = 4$, so the line crosses the y-axis at the point $(0, 4)$.

Plotting two points:

• At $x = 0$, $y = 4$ (point $(0, 4)$).
• At $x = 1$, $y = -3(1) + 4 = 1$ (point $(1, 1)$).

Draw the straight line through these points.

Special Cases:

• Horizontal Line: If $m = 0$, the equation becomes $y = b$, which means the line is horizontal and crosses the y-axis at $y = b$.
• Vertical Line: If the equation is in the form $x = a$, it represents a vertical line passing through $x = a$.

Summary:

• A linear graph represents a straight-line relationship between two variables.
• The graph of a linear equation can be plotted by identifying the slope and y-intercept and then drawing a straight line through two points.