Gradients

The graph helps to properly analyze and give a correct interpretation of given information. It is therefore important to learn graphing fundamentals.

Gradients
Gradients state how steep a line is. It’s also called the slope. A line may have a positive or a negative gradient. Positive gradient means the slope is going upward. Negative gradient means the slope is going downward.

Steps in Finding the Gradient of a Straight Line:
a. Choose any two different points on a line. Get the coordinates of the points.
b. From the two different points, make a right-angled triangle with the line connecting the two points being the longest line or hypotenuse.
c. Divide the length of the vertical line to the horizontal line to get the gradient.

Example:

Let’s find the slope of this line. Choose any points within the line and get their coordinates. Let’s have (4, 5) and (10,11).

Solution 1
When connecting the two points, can make a right triangle. The vertical length is (10,11) and (10, 5). The horizontal length is (4, 5) and (10,5).

For the length of vertical and horizontal lines:
a. Vertical length is (10,11) and (10, 5) is 6 units.
b. Horizontal length is (4, 5) and (10,5) is 6 units.
c. Compute the gradient.

Vertical length ÷ horizontal length = 6/6 = 1. The gradient is 1.

Solution 2
(4, 5) and (10,11)
(4 – 10) / (5 – 11) = (- 6) / (- 6) = 1

Since the line crosses the y-axis at y = 1, then the equation for this graph is y = x + 1.

Finding the Gradient of a Curve Graph
a. Choose any point on the curve.
b. From the point, draw a tangent line. A tangent line is a line that passes through the curve and has the same gradient.
c. At the point on a curve and a tangent line, draw a right triangle.
d. Compute for the gradient of the line.

Example:

The blue line is the tangent line that passes through the point (0.75, 0.25). The point on the tangent line is (3, 4).

(0.75 – 3) / (0.25 – 4) = (-2.25) / (-3.75) = 0.6

The gradient of the curve is approximately 0.6.

Parallel Lines
Note: Parallel lines have the same gradient.

Example:

Compute the gradient.

a. Point (-1, 2) and (0, 4).
(-1 – 0 ) / ( 2 – 4) = (- 1) / (- 2) = 0.5

b. Point (1, 2) and (2, 4).
(1 – 2 ) / ( 2 – 4) = (- 1) / (- 2) = 0.5

The two lines have the same gradient, which is 0.5.

Perpendicular Lines
Note: The product of the gradient of a perpendicular line is equal to -1.

Example:

Find the gradient of the perpendicular line given the equation of the lines y = 2x and y = -½ x through point (0, 0).
First line: m = 2
Second line: m = -½
If we multiply the gradient of the two, 2 (-½ ) = -1.