The Slope of a Line Revision Notes for Leaving Cert Mathematics

The Slope of a Line

What is slope?

Slope (also called gradient) measures how steep a line is. It tells us how much the y-value changes when the x-value increases by 1 unit.

In mathematics, we use slope to study rates of change - how one quantity varies with respect to another. This is a fundamental concept in calculus.

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Key definition: The slope of a line is the rate at which y changes with respect to x.

Calculating slope using rise over run

The most common way to find slope is using the rise over run method.

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Formula: Slope = $\frac{\text{rise}}{\text{run}} = \frac{\text{change in y}}{\text{change in x}}$

This diagram shows how we can use a right triangle to find the slope. When we move 1 unit right (run = 1) and 2 units up (rise = 2), the slope is $\frac{2}{1} = 2$ .

Finding slope from linear equations

When a line is written in the form $y = mx + b$ :

m is the slope
b is the y-intercept

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Worked Example 1: Finding slope from y = 2x

For the line $y = 2x$ :

The slope is 2
This means for every 1 unit increase in x, y increases by 2 units

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Worked Example 2: Interpreting fractional slope

Consider the line $y = \frac{2}{3}x$ :

The slope is $\frac{2}{3}$ , which means:

Rise = 2 units
Run = 3 units
Slope = $\frac{2}{3}$

First differences and slope

For linear functions, we can find the slope by examining the first differences in a table of values.

This table shows that when x increases by 1, y always increases by 2. The first difference is constant at +2, which equals the slope.

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Important rule: For linear functions, slope = first difference

Mathematical notation for slope

In calculus, we write slope using the notation $\frac{dy}{dx}$ (read as "dy by dx").

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Key relationship: $\frac{dy}{dx} = \text{slope} = \text{first difference}$ (for linear functions)

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Worked Example 3: Slope in calculus notation

For different linear equations:

$y = 4x$ → $\frac{dy}{dx} = 4$
$y = -3x + 5$ → $\frac{dy}{dx} = -3$
$y = \frac{1}{2}x + 6$ → $\frac{dy}{dx} = \frac{1}{2}$

General rule: For $y = axe + b$ , the slope $\frac{dy}{dx} = a$

Types of slopes

Positive slope

Lines that go upward from left to right have positive slopes.

Graph (i) shows a line with positive slope - as x increases, y also increases.

Negative slope

Lines that go downward from left to right have negative slopes.

Graph (ii) shows a line with negative slope - as x increases, y decreases.

Zero slope

Horizontal lines have zero slope because y doesn't change as x changes.

For the horizontal line $y = 6$ :

The slope is 0
$\frac{dy}{dx} = 0$ because there's no change in y

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Rule: If $y = c$ (where c is a constant), then $\frac{dy}{dx} = 0$

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Worked Example 4: Reading slopes from a graph

Looking at this piecewise function:

From A to B: positive slope (line goes up)
From B to C: zero slope (horizontal line)
From C to D: negative slope (line goes down)
From D to E: negative slope (line continues down)

Summary of key formulas

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Essential Formulas:

Basic slope formula: Slope = $\frac{\text{rise}}{\text{run}}$
From linear equation: If $y = mx + b$ , then slope = $m$
Calculus notation: Slope = $\frac{dy}{dx}$
Constant function: If $y = k$ , then slope = $0$

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Key Points to Remember:

Slope measures the steepness of a line - it tells us the rate of change of y with respect to x
Use "rise over run" to calculate slope from a graph or two points
In linear equations $y = mx + b$ , the coefficient m is always the slope
Positive slopes go up, negative slopes go down, and horizontal lines have zero slope
For linear functions, the slope equals the first difference and can be written as $\frac{dy}{dx}$

The Slope of a Line (Leaving Cert Mathematics): Revision Notes