Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Find the equation of a line given two points.
• Find the equation of a line given the gradient and one point.
• Find the equation of a line given the gradient and y-intercept.
• Calculate the angle of inclination of a line.
• Determine if lines are parallel or perpendicular.

Lesson summary

Analytical geometry is a powerful tool that combines algebra and geometry. It allows us to describe geometric shapes and relationships using algebraic equations. This is crucial for understanding concepts in physics (e.g., projectile motion), engineering (e.g., structural design), and even computer graphics. In South Africa, understanding analytical geometry is vital for careers in infrastructure development, urban planning, and resource management. We'll be focusing on key skills this week to master the equations of straight lines.

Lesson notes

2. 1. The Equation of a Straight Line The most common form of the equation of a straight line is the gradient-intercept form: `y = mx + c` Where: `y` is the y-coordinate of any point on the line. `x` is the x-coordinate of any point on the line. `m` is the gradient (slope) of the line. It represents the steepness and direction of the line. A positive `m` indicates an increasing line (from left to right), while a negative `m` indicates a decreasing line. The gradient can be calculated using the formula: `m = (y2 - y1) / (x2 - x1)` where (x1, y1) and (x2, y2) are two distinct points on the line. `c` is the y-intercept, which is the y-coordinate of the point where the line crosses the y-axis (i.e., where x = 0). 2.

2. Finding the Equation of a Line There are several ways to find the equation of a straight line, depending on the information you're given: Given the gradient (m) and the y-intercept (c): Simply substitute the values of `m` and `c` into the equation `y = mx + c`. Given the gradient (m) and a point (x1, y1): Use the point-slope form: `y - y1 = m(x - x1)` Substitute the values of `m`, `x1`, and `y1` into this equation, and then rearrange it into the gradient-intercept form (y = mx + c). This makes it easier to visually identify the gradient and y-intercept. Given two points (x1, y1) and (x2, y2): First, calculate the gradient `m` using the formula: `m = (y2 - y1) / (x2 - x1)`. Then, use either the point-slope form `y - y1 = m(x - x1)` or `y - y2 = m(x - x2)`. Both will give you the same answer. Substitute the gradient `m` and the coordinates of one of the points into the chosen equation, and rearrange to get the gradient-intercept form. 2.

3. Angle of Inclination The angle of inclination (θ) of a straight line is the angle it makes with the positive x-axis, measured counter-clockwise. The angle of inclination is related to the gradient by the formula: `m = tan θ` Therefore, to find the angle of inclination, calculate the gradient `m` and then find the inverse tangent (arctan or tan -1 ) of `m`: `θ = arctan(m)` Make sure your calculator is in degree mode. 2.

4. Parallel and Perpendicular Lines Parallel Lines: Two lines are parallel if and only if they have the same gradient. If line 1 has gradient `m1` and line 2 has gradient `m2`, then the lines are parallel if `m1 = m2`.

Perpendicular Lines: Two lines are perpendicular if and only if the product of their gradients is -

1. If line 1 has gradient `m1` and line 2 has gradient `m2`, then the lines are perpendicular if `m1 m2 = -1`. This can also be written as `m2 = -1/m1`. This means that the gradient of the second line is the negative reciprocal of the gradient of the first line.

Worked example

Example 1: Find the equation of the line that passes through the points (1, 5) and (3, 11).

Calculate the gradient:

`m = (y2 - y1) / (x2 - x1) = (11 - 5) / (3 - 1) = 6 / 2 = 3`

Use the point-slope form (using point (1, 5)):

`y - 5 = 3(x - 1)`

Rearrange to gradient-intercept form:

`y - 5 = 3x - 3`

`y = 3x + 2`

Therefore, the equation of the line is `y = 3x + 2`.

Example 2: Find the equation of the line with a gradient of -2 that passes through the point (-1, 4).

Use the point-slope form:

`y - 4 = -2(x - (-1))`

`y - 4 = -2(x + 1)`

Rearrange to gradient-intercept form:

`y - 4 = -2x - 2`

`y = -2x + 2`

Therefore, the equation of the line is `y = -2x + 2`.