Graphing Lines (Leaving Cert Mathematics): Revision Notes

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:

$$axe + by + c = 0$$

Steps to Graph a Line

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

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

3. Plot the Y-Intercept: Start by marking the y-intercept on the graph.
4. Use the Slope: From the y-intercept, use the slope $m = \frac{\text{rise}}{\text{run}}$ to find additional points.
5. 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 $x-coordinate$.

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Worked Examples

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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.