Slope

In this section, we will explore what slope means, how to interpret it, and how to calculate it using the rise over run method and the slope formula. Let's break it down step by step so it's easy to understand.

What is Slope?

Slope is a measure of how steep a line is on the Cartesian Plane. It tells us how much the line rises or falls as we move from left to right. The slope can be:

Positive: The line goes up as you move from left to right.
Negative: The line goes down as you move from left to right. You can think of slope as how much you " $rise$ " vertically for each step you " $run$ " horizontally.

Rise Over Run Method

One way to think about slope is with the rise over run method:

Rise: This is how much the line goes up or down (vertical change).
Run: This is how much the line goes left or right (horizontal change). The slope is calculated as:

$\text{Slope} = \frac{\text{Rise}}{\text{Run}}$

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Example 1: Rise Over Run Let's say we have a line on the graph, and we want to find its slope by looking at how much it rises and runs between two points.

Step 1: Pick two points on the line. Let's say one point is $(1, 2)$ and the other point is $(4, 5)$ .

Step 2: Count how much you move up ( $rise$ ) and how much you move right ( $run$ ) to get from the first point to the second.
Rise: From $y = 2$ to $y = 5$ , you move up 3 units.
Run: From $x = 1$ to $x = 4$ , you move right 3 units. So, the slope is:

$\text{Slope} = \frac{\text{Rise}}{\text{Run}} = \frac{3}{3} = 1$

This means the line goes up 1 unit for every 1 unit it goes to the right.

Slope Formula

The slope can also be calculated using the slope formula when you know the coordinates of two points on the line. The formula is:

$\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}$

Here's how you can use it:

Label the Points:

Let's say you have two points, $A$ and $B$ , with coordinates $A(x_1, y_1)$ and $B(x_2, y_2)$ .
$x_1$ and $y_1$ are the coordinates of the first point, and $x_2$ and $y_2$ are the coordinates of the second point.

Substitute into the Formula:

Plug the values from the points into the formula. Make sure to put the numbers in brackets, especially if they are negative.

Calculate:

Do the math to find the slope. This tells you how steep the line is.

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Example 2: Finding Slope Using the Formula Let's find the slope of the line passing through the points $A(2, 3)$ and $B(6, 11)$ .

Step 1: Label the points:

For point $A$ , $x_1 = 2$ and $y_1 = 3$ .
For point $B$ , $x_2 = 6$ and $y_2 = 11$ .

Step 2: Substitute into the formula:

Start with the slope formula: $\text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}$
Substitute the coordinates: $\text{Slope} = \frac{(11) - (3)}{(6) - (2)}$

Step 3: Calculate:

Subtract the $y-values$ : $11 - 3 = 8$ .
Subtract the $x-values$ : $6 - 2 = 4$ .
Divide to find the slope: $\text{Slope} = \frac{8}{4} = 2$

This means the line rises 2 units for every 1 unit it runs to the right.

What Does the Slope Tell Us?

Positive Slope: If the slope is positive, the line goes up as you move from left to right.
Negative Slope: If the slope is negative, the line goes down as you move from left to right.
Zero Slope: If the slope is zero, the line is perfectly horizontal.
Undefined Slope: If the slope is undefined (division by zero), the line is vertical.

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Key Tips for Success

Always Label: Start by labeling your points as $x_1$ , $y_1$ and $x_2$ , $y_2$ to avoid mixing up the coordinates.
Substitute Carefully: Use brackets when substituting values, especially with negative numbers, to avoid errors.
Understand the Slope: Remember that slope tells you how steep a line is and in which direction it goes.

Slope is an important concept in co-ordinate geometry, and understanding it will help you with many other topics in mathematics. Keep practicing, and it will soon become second nature!

Slope Simplified Revision Notes for Junior Cycle Mathematics

Slope

What is Slope?

Rise Over Run Method

Slope Formula

What Does the Slope Tell Us?

Key Tips for Success

500K+ Students Use These Powerful Tools to Master Slope For their Junior Cycle Exams.

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