Coordinate Geometry

Grade 12 Math - Coordinate Geometry

Lesson Objectives

Identify and interpret gradients (slopes) and intercepts from equations.
Rewrite linear equations in slope-intercept and general forms.
Plot linear equations using gradient and intercepts.
Solve problems involving lines and their intersections.

Lesson Introduction

Coordinate Geometry connects algebra and geometry using graphs. Linear equations in two variables represent straight lines, and key features include the gradient (slope), y-intercept, and x-intercept. This lesson explores how to identify these components and graph lines effectively in the Cartesian plane.

Core Lesson Content

General Form and Slope-Intercept Form

A linear equation in two variables is usually written as:

General form: $Ax + By + C = 0$
Slope-intercept form: $y = mx + c$ , where $m$ is the gradient and $c$ is the y-intercept.

Finding the Gradient (Slope)

The gradient of a straight line is given by:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

It represents the rate of change of $y$ with respect to $x$ .

Finding the Intercepts

Y-intercept: Set $x = 0$ and solve for $y$ .
X-intercept: Set $y = 0$ and solve for $x$ .

Graphing Linear Equations

Use either intercepts, a table of values, or gradient and intercept to sketch the graph.

Worked Examples

Example 1: Find the gradient and intercepts of

y = 2x + 3

Gradient:

m = 2

, y-intercept:

(0, 3)

, x-intercept:

x = -\frac{3}{2}

Example 2: Convert

3x + 2y - 6 = 0

to slope-intercept form and find

m

and

c

2y = -3x + 6 \Rightarrow y = -\frac{3}{2}x + 3

Example 3: Find the equation of the line passing through (2, 5) and (6, 13)

m = \frac{13 - 5}{6 - 2} = 2

, then use point-slope:

y - 5 = 2(x - 2)

⇒

y = 2x + 1

Example 4: Graph

y = -x + 4

Use points: (0, 4), (1, 3), (2, 2), draw a straight line

Example 5: Determine the x- and y-intercepts of

4x - y = 8

y-intercept: set

x = 0

-y = 8 \Rightarrow y = -8

; x-intercept:

y = 0 \Rightarrow 4x = 8 \Rightarrow x = 2

Example 6: Find the gradient of a line perpendicular to

y = \frac{1}{3}x - 2

Perpendicular gradient:

-3

Example 7: Find the equation of a line with gradient -2 passing through (1, -3)

y + 3 = -2(x - 1) \Rightarrow y = -2x - 1

Example 8: Determine if points A(1, 2), B(3, 6), C(5, 10) lie on a straight line.
Find gradient AB and BC: both

m = 2

, so they lie on a straight line.

Example 9: Determine the gradient of the line

5x + y = 10

y = -5x + 10 \Rightarrow m = -5

Example 10: Sketch the line

x = 3

Vertical line through

x = 3

; undefined gradient.

Exercises

Find the gradient and intercepts of $y = -3x + 2$
[WAEC] Convert $2x - y = 5$ to slope-intercept form and state its gradient [Past Question]
[NECO] Find the equation of the line through (4, -2) with gradient $\frac{1}{2}$ [Past Question]
Graph $y = 0.5x - 1$ using a table of values
Determine x- and y-intercepts of $6x + 3y = 12$
[NECO] Find the equation of the line passing through (1, 1) and (4, 10) [Past Question]
Find the gradient of a line perpendicular to $y = -2x + 4$
Determine whether points (0, 0), (2, 4), and (3, 6) lie on a straight line
Sketch the line $y = 4$
[WAEC] Find the x-intercept of $y = 5x - 10$ [Past Question]

Conclusion/Recap

Understanding the structure of linear equations helps in interpreting and constructing graphs. Identifying gradients and intercepts is essential for solving real-life problems, from physics to economics. Mastery comes through regular practice and applying various forms of linear equations.