Planning tool

Students will:

• vary the parameters of graphs of related functions
• use digital tools for easy repetition and manipulation
• make conjectures, test and identify emerging patterns and trends
• visualise linear functions
• restrict intervals.

Students may:

• be uncomfortable with plotting linear equations and identifying emerging patterns.
• be able to operate digital tools easily but fail to make connections and find patterns both graphically and algebraically.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention

• We are learning how to adjust the values of a linear function to create any straight line we want.
• We are learning how restricting the values for or adjusts the length of the straight line. 

Why are we learning about this?

Visualisation is key to solving complex problems. Mathematical equations are merely symbolic representations of lines, curves, shapes or solids. Far more information is contained in the image than in the equation itself. Developing the ability to intuit what an equation’s graph looks like makes it much easier to solve related problems and make sense of solutions.

What to do

Write your name in linear graphs!

Using the GeoGebra calculator, we will create a series of graphs that join to write our initials or name, or to draw a picture. The more you practise and the better you understand the role of each value in a linear function, the more complex your output can be.

Let’s quickly review the basics and consider:  y = mx + c

m controls the slope of the line. If it is a positive number, the line will slope up. If it is a negative number, the line will slope down. Remember, we always think in terms of left-to-right. As m increases in size, the steepness of the line increases too.

c controls the point at which the line crosses the y-axis. Whatever value c takes is the point at which it intercepts the y-axis. Therefore, increasing or decreasing this value will move the graph up or down.

Write my name

To write my initials I need to create an S and a G. I will begin with the S as an example to get you started. To challenge myself I’ve decided to make a funky slanted S. It is made up of three straight line graphs.

1. First I will make Line 1 (green line below, f), and then I’ll use this as the basis for the other two lines. Line 2 (blue line, g) is perpendicular to Line 1. Hint: when the gradients (m) of perpendicular lines are multiplied together, the result is always –1. So, if I made the gradient of Line 1 equal to 3, I’ll need to set the gradient of Line 2 to –1  3.
2. I need to shift Line 2 down, which I will do by adjusting the c value until it is in the exact spot I want.
3. Line 3 (purple line, h) should have an identical gradient to Line 1, but it will also need to be shifted down.

You can see Line 1 and Line 3 are parallel and perpendicular to Line 2 in the example above. But it is difficult to see the S! There’s one last step.

The lines currently go off to infinity in both directions. They need to be shortened. I will do this by restricting the values in each function. I’ll need to know where the lines intersect first. GeoGebra can help with that. Simply click on the point of intersection.

I need to restrict the x values of Line 2 to between x=–6.4 and x=–4.4 only. The x-values of Line 1 can begin at x=–6.4, and the x-values of Line 3 (h) must stop at x=–4.4 I’ll play around with the other end points until I am happy with the look of my S.

To restrict the x-values in GeoGebra we need to use a ‘function’. Type function into the input space and select the second option. You will need to type in the name of the line you want to restrict, the x value you want it to begin at, and the x value you want it to end at.

E.g. Function(g,–6.4,–4.4)

Lastly, uncheck the original graph of ‘g’ so you can only see the restricted version. Now I can move on to the G in my initials.

Start with your initials, then try your first name and finally attempt to draw something.

Success criteria

• I can adjust the m and c values in a linear function to create my desired straight line.
• I can restrict the domain of a function to select only the piece of the line required.