Slope

The ten percent slope in the road signs above mean that if you move 100 m along the horizontal direction, then you move at an average 10 m along the vertical direction.

In order to find the percentage slope, you divide the movement along the y-axis with the movement along the x-axis, \(\frac{10}{100}=0.1=10\)%.

The road signs above assume that you drive from the left to the right, from the look of the pictures you can deduce whether it is a slope downwards or upwards.

When defining slope in mathematics, the definition is similar to the slopes on the road signs. The slope is defined to be the difference along the y-axis divided by the difference along the x-axis. In order to distinguish between a slope downwards and a slope upwards, the difference along the y-axis is counted as negative if the slope is downwards when going from the left to the right.

Instead of the words "difference along", the symbol \(\Delta \) is used. The difference along the y-axis is denoted by \(\Delta y\) and the difference along the x-axis by \(\Delta x\). If \(m\) denotes the slope, we get following formula: \[m=\frac{\Delta y}{\Delta x}\]

Exercises

Exercise 1 - Slope by definition

Start a new GeoGebra sheet. Right-click on the drawing pad and check both Axes and Grid.

Choose Options->Point Capturing->On (Grid).

Use the tool Line through Two Points to make four lines through the points:

1. (-1,1) and (3,4)
2. (0,-1) and (4,-1)
3. (2,-2) and (-2,2)
4. (1,1) and (1,4)

Find the slope of the four lines by using the definition \(m=\dfrac{\Delta y}{\Delta x}\) and write down your answers!

Exercise 2 - Using the tool Slope

Use the tool Slope on the four lines. Click on the tool and then on a line.

Compare the GeoGebra-slopes to your answers.

Comment

The GeoGebra-way of defining slope could be seen as an alternative definition: