Functions as Machines

Here you will find advanced tasks for function machines. In particular, parameters and the idea of substitution are new additions.

1 Parameters

In Math-Nodes, function machines have not only one input for the independent variable. With the additional inputs, you can use parameters.

Simply specify the name of the connected parameter in the function equation.

Connect the parameter \(a\) from the example to the function machine and change the set value. Describe how the graph of the function changes depending on the parameter value.

2 Hit the Graph

In the graph card, you can see the graph of a quadratic function. Set the parameters of function \(f\) so that the graph of \(f\) matches the white one. Give the set parameter values in the workbook and describe what each of the 3 parameters changes in the graph.

Find as many quadratic functions/parameter settings as possible so that \( f(1)=4\). Sketch 3 solutions that are as different as possible. Give the parameter values.

Substitution

At the parameter inputs of a function machine, you can also connect other functions. To do this, the machine name and the independent variable must be entered in the form \(f(t)\) into the other function. This way, you can make your patch clearer and, for example, compare the influence of different machines. In the following task, you will see an example.

3 Where is the Parabola?

In the functions \(g\) and \(h\), the function \(a\) has been used in different places. Assign the functions to their graphs and justify your assignment.

Draw the square function \(a\) into the graphs in the workbook.

4 Symmetrical or Not?

You always see pairs of functions here, where one is inserted into the other. The second function is always the same. Match the pairs to the graphs. Justify your assignment in the workbook.

5 Symmetry Change

Choose the functions \(a\) so that the overall functions are axisymmetric. You can enter the function freely, but you must keep the function name (\(a(x)=\:\)).

Find as many function types as possible and specify them in the workbook.

    6 Function Puzzle with Parameters

    Link using operations and/or compose the machines and connect and set the parameters so that the white graphs are created. For each case, give the corresponding function equation and parameter values in the workbook.

    If you have solved it correctly, no card is left over.