Functions as Machines

Here you will find advanced tasks for function machines. In particular, parameters and the idea of modulation 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.

3 Modulation

First consider the functions \(f\) and \(g\) in the first window. What influence do parameters \(a\) and \(b\) each have on the graph?

a) Try it out and describe in the workbook.

b) The functions \(f\) and \(g\) in the second patch look almost like the first one. However, instead of parameters, functions \(a\) and \(b\) are connected to the parameter inputs here. Describe the course of the graphs of \(f\) and \(g\) and the influence of functions \(a\) and \(b\) on the course.

Tip: Your knowledge about the influence of parameters can help you.

Tip

You can imagine parameters as knobs with which you can change the function. You can ask yourself questions like: "What does it do to the function when this parameter is very large or close to zero?"

Sometimes it's helpful to imagine parts of a function as if they were replacing one of these knobs. For \(g(x)=sin(b(x)\cdot x)\), you can imagine that someone moves the knob the way the connected function \(b(x)\) runs. Imagine the parameter \(b\) is turned back and forth or turned up faster and faster. If you know what a parameter would influence at the position of function \(g\), you also know what function \(b\) does there.

4 How Does It Run?

Functions \(f\) and \(g\) are each modulated with functions \(a\) and \(b\). Formulate hypotheses in the workbook about how the courses of \(f\) and \(g\) differ without displaying the graphs of \(f\) and \(g\). Then check your hypotheses.

Tip: You may look at the graphs of \(a\) and \(b\).

5 Modulation Detective

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

6 What Was Modulated Here?

In functions \(g\) and \(h\), different parameters have been modulated by the same function. Match the functions to their graphs and justify your assignment.

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

7 Symmetry Change

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

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

    8 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. Give the corresponding function equation and parameter values in the workbook for each.

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