Graphing Lines

What is Graphing a Line?

Graphing a line involves plotting all the points that satisfy the equation of the line in a Cartesian plane. The equation of a line can take various forms, such as:

Slope-Intercept Form:

y = mx + c

$m$ is the slope.
$c$ is the $y$ -intercept.

Point-Slope Form:

y - y_1 = m(x - x_1)

Used when the slope and a point on the line are known.

General Form:

ax + by + c = 0

Steps to Graph a Line

Rewrite the Equation: Convert the equation to slope-intercept form, $y = mx + c$ , if necessary.
Identify Key Features:

Slope ( $m$ ): Rise over run, indicating steepness and direction.
Y-Intercept ( $c$ ): The point where the line crosses the $y$ -axis.

Plot the Y-Intercept: Start by marking the y-intercept on the graph.
Use the Slope: From the y-intercept, use the slope $m = \frac{\text{rise}}{\text{run}}$ to find additional points.
Draw the Line: Connect the points with a straight line extending in both directions.

Horizontal and Vertical Lines

Horizontal Line ( $y = k$ )****: All points have the same $y$ -coordinate.
Vertical Line ( $x = k$ )****: All points have the same $x$ -coordinate.

Worked Examples

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Example 1:

Graph $y = 2x + 1$

Identify the $y$ -intercept ( $c$ ): $(0, 1)$
Identify the slope ( $m$ ): $2 = \frac{2}{1}$ (rise = $2$ , run = $1$ ).
Plot the $y$ -intercept $(0, 1)$
From $(0, 1)$ , move $2$ units up and $1$ unit right to find another point $(1, 3)$
Draw the line through $(0, 1)$ and $(1, 3)$

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Example 2:

Graph $2x-y=4$

Rewrite the equation in slope-intercept form:

y = 2x - 4

Slope ( $m$ ): $2$
Y-intercept ( $c$ ): $-4$

Plot the $y$ -intercept $(0, -4)$
From $(0, -4)$ , use the slope $2 = \frac{2}{1}$ to find another point:

Move 2 units up and 1 unit right to $(1, -2)$

Draw the line through $(0, -4)$ and $(1, -2)$

Summary

Graphing Steps: Rewrite the equation, identify the slope and intercept, plot key points, and draw the line.
Forms of Line Equations: 10. $y = mx + c$ (slope-intercept form). 11. $y - y_1 = m(x - x_1)$ (point-slope form). 12. $ax + by + c = 0$ (general form).
Special Lines:
- Horizontal line ( $y = k$ ): Parallel to $x$ -axis.
- Vertical line ( $x = k$ ): Parallel to $y$ -axis.
Practice graphing lines to improve understanding and precision.

Graphing Lines Simplified Revision Notes for Leaving Cert Mathematics