The Gradient of a Straight Line Revision Notes for VCE SSCE Mathematical Methods

The Gradient of a Straight Line

What is a gradient?

The gradient of a straight line is a measure of its steepness or slope. It tells us how much the line rises (or falls) for every unit we move horizontally.

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An important fact to remember: through any two points, you can only draw one unique straight line. This means a straight line can be completely defined by knowing just two points on it.

Understanding rise and run

To understand gradient, we need to know two key terms:

Rise: The vertical change in $y$ -values as you move from one point to another on the line.

Run: The horizontal change in $x$ -values as you move from one point to another on the line.

The gradient is the ratio of rise to run:

\text{Gradient} = \frac{\text{rise}}{\text{run}}

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The gradient tells us how steep a line is by comparing how much it rises vertically to how much it runs horizontally. A larger gradient means a steeper line!

The gradient formula

For any two points on a line, $A(x_1, y_1)$ and $B(x_2, y_2)$ , the gradient $m$ can be calculated using:

m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}

chatImportant

It doesn't matter which point you label as the first point and which as the second. This is because:

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

Both expressions give you the same gradient value.

Properties of gradients

Understanding what different gradient values mean:

Positive gradient: When a line slopes upward from left to right, the gradient is positive. This happens when both the rise and run have the same sign (both positive or both negative).

Negative gradient: When a line slopes downward from left to right, the gradient is negative. This happens when the rise and run have opposite signs.

Zero gradient: A horizontal line (parallel to the $x$ -axis) has a gradient of zero. This is because $y_2 - y_1 = 0$ , so the rise is zero.

Undefined gradient: A vertical line (parallel to the $y$ -axis) has an undefined gradient. This is because $x_2 - x_1 = 0$ , and we cannot divide by zero.

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Think of gradient as describing the direction of a line:

Positive gradients go "uphill" from left to right
Negative gradients go "downhill" from left to right
Zero gradients are flat (horizontal)
Undefined gradients are completely vertical

Worked example: Finding gradient from a graph

Find the gradient of each line:

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Worked Example: Finding Gradient from a Graph

Part a:

Let $(x_1, y_1) = (-2, 0)$ and $(x_2, y_2) = (0, 2)$

Using the gradient formula:

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

m = \frac{2 - 0}{0 - (-2)}

m = \frac{2}{2}

m = 1

The gradient is $1$ . Notice this line slopes upward from left to right, giving a positive gradient.

Part b:

Let $(x_1, y_1) = (0, 3)$ and $(x_2, y_2) = (2, 0)$

Using the gradient formula:

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

m = \frac{0 - 3}{2 - 0}

m = -\frac{3}{2}

The gradient is $-\frac{3}{2}$ . This line slopes downward from left to right, giving a negative gradient.

Worked example: Finding gradient from coordinates

Find the gradient of the line that passes through the points $(1, 6)$ and $(-3, 7)$ .

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Worked Example: Finding Gradient from Coordinates

Solution:

Using the gradient formula:

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

m = \frac{7 - 6}{-3 - 1}

m = \frac{1}{-4}

m = -\frac{1}{4}

Alternative approach: We could also calculate using:

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

m = \frac{6 - 7}{1 - (-3)}

m = \frac{-1}{4}

m = -\frac{1}{4}

Both methods give the same answer: $m = -\frac{1}{4}$ .

The angle of slope

The gradient of a line is closely related to the angle the line makes with the positive $x$ -axis.

Positive gradients and acute angles

When a line has a positive gradient, it forms an acute angle $\theta$ (less than $90°$ ) with the positive direction of the $x$ -axis.

First, recall the trigonometric ratio for tangent from Year 10:

\tan \theta = \frac{\text{opposite}}{\text{adjacent}}

For a line with positive gradient, the relationship between gradient and angle is:

m = \tan \theta

where $\theta$ is the acute angle the line makes with the positive $x$ -axis.

From the diagram, we can see that:

m = \frac{y_2 - y_1}{x_2 - x_1} = \tan \theta

chatImportant

The key relationship to remember: gradient equals the tangent of the angle.

m = \tan \theta

This formula connects the algebraic concept of gradient with the geometric concept of angle.

Worked example: Finding gradient and angle (positive gradient)

Determine the gradient of the line passing through the points $(3, 2)$ and $(5, 7)$ and the angle $\theta$ that the line makes with the positive direction of the $x$ -axis.

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Worked Example: Finding Gradient and Angle (Positive Gradient)

Solution:

First, find the gradient:

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

m = \frac{7 - 2}{5 - 3}

m = \frac{5}{2}

Since the gradient is positive, the angle $\theta$ is acute. Using the relationship $m = \tan \theta$ :

\tan \theta = \frac{5}{2}

\theta = \tan^{-1}\left(\frac{5}{2}\right)

\theta = 68.1986...°

\theta = 68.20°

(correct to two decimal places)

Exam tip: Use your calculator's inverse tangent function to find the angle. Make sure your calculator is in degree mode.

Negative gradients and obtuse angles

When a line has a negative gradient, it forms an obtuse angle $\theta$ (greater than $90°$ but less than $180°$ ) with the positive direction of the $x$ -axis.

The line forms an acute angle $\alpha$ with the negative direction of the $x$ -axis, and we have:

\theta = 180° - \alpha

The gradient can be expressed as:

m = -\tan \alpha

Using the tangent identity for supplementary angles:

\tan \theta = \tan(180° - \alpha) = -\tan \alpha

Therefore:

m = \tan \theta

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This formula $m = \tan \theta$ works for both positive and negative gradients. The difference is just in how we interpret the angle:

Positive gradient → acute angle (less than $90°$ )
Negative gradient → obtuse angle (between $90°$ and $180°$ )

Worked example: Finding gradient and angle (negative gradient)

Determine the gradient of the line passing through the points $(5, -3)$ and $(-1, 5)$ and the angle $\theta$ that the line makes with the positive direction of the $x$ -axis.

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Worked Example: Finding Gradient and Angle (Negative Gradient)

Solution:

First, find the gradient:

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

m = \frac{5 - (-3)}{-1 - 5}

m = \frac{8}{-6}

m = -\frac{4}{3}

Since the gradient is negative, the angle $\theta$ is obtuse.

\tan \theta = -\frac{4}{3}

To find $\theta$ , first find the acute angle $\alpha$ using:

\tan \alpha = \frac{4}{3}

\alpha = \tan^{-1}\left(\frac{4}{3}\right)

\alpha = 53.130...°

Now find the obtuse angle $\theta$ :

\theta = 180° - \alpha

\theta = 180° - 53.130...°

\theta = 126.87°

(correct to two decimal places)

Exam tip: For negative gradients, use your calculator to find the inverse tangent of the positive value (without the negative sign), then subtract this from $180°$ to get the obtuse angle.

Remember!

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Key Points to Remember:

The gradient formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are any two points on the line.
Lines sloping upward from left to right have positive gradients; lines sloping downward from left to right have negative gradients.
Horizontal lines have a gradient of zero (no vertical change), while vertical lines have undefined gradient (no horizontal change, division by zero).
The gradient equals the tangent of the angle with the positive $x$ -axis: $m = \tan \theta$ .
For positive gradients, $\theta$ is acute (less than $90°$ ). For negative gradients, $\theta$ is obtuse (between $90°$ and $180°$ ), and you calculate it using $\theta = 180° - \alpha$ where $\alpha$ is the acute angle found from the positive gradient value.

The Gradient of a Straight Line (VCE SSCE Mathematical Methods): Revision Notes