Linear Functions

Slope

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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.

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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.

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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 icon 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 icon 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:

If the change along the x-axes (moving from left to right) is 1,
then the change along the y-axes is the slope.

 

Slope

Exercise 3

Start a new GeoGebra sheet. Make sure that the input bar is shown; if not, pick View->Input Bar.

Enter following three functions in the input bar (one at the time):

  • \(y=x\)
  • \(y=x+2\)
  • \(y=x-1\)

Without plotting the graph, answer these questions about the function \(y=x+4\).

  1. What is the slope?
  2. What is the y-intercept?
  3. What is the x-intercept?

Exercise 4

Delete the three functions and instead enter following functions in the input bar:

  • \(y=-2x\)
  • \(y=-2x+3\)
  • \(y=-2x-2\)

Without plotting the graph, answer these questions about the function \(y=-2x+1\).

  1. What is the slope?
  2. What is the y-intercept?
  3. What is the x-intercept?

Exercise 5

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Delete the three functions and enter the equation \(y=2x\) in the input bar.

The equation is shown in the algebra view and the line is shown in the drawing pad.

Use the mouse to drag the line and watch the equation in the algebra view. The second number shown in the expression, changes as you move the graph. Describe how you can find the second number of the expression by looking at the graph!

Exercise 6

Without drawing a line, answer these questions about the line defined by \(y=3x-2\).

  1. What is the slope?
  2. What is the y-intercept?

Comment

GeoGebra distinguishes between functions and equations. Functions are written like this: \(f(x), f1(x), g(x), MalinsFunction(x),..\). Equations are written using the variables \(x\) and \(y\). Each equation is given a unique name, but all equations use the variables \(x\) and \(y\).

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An equation can be used to define a line like this:

Only the points whose coordinates x and y satisfy the equation
(makes the equality true),
lie on the line

A longer explanation is given on next page.

Gradients of Perpendicular Lines

  • Draw a line through points A and B.
  • Draw a line perpendicular to the first line through the point A.
  • Use the IconSlope tool on both lines.
  • Rename the gradients to m1 and m2 respectively.
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  • Make a new variable as in the picture below. Move the points!
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  • Make a conjecture! Prove it!

Hint: Look at the picture and use similar triangle.

reference:

the pictures of the road signs are from: http://commons.wikimedia.org/wiki/Category:Slope_signs

by Malin Christersson under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Sweden License

www.malinc.se