Create a new figure ith the menu item *File - New Figure with frame - Without vectors* and choose an orthonormal frame.

Create six points of coordinates (-2;0), (-1;1), (0;0), (1;-1), (2;0) et (4;2) (tool ).

Create six calculus named *m1*, *m2*, ..., *m6* and with formulas respectively 2, 0, -2, 0, 2 et 2 (tool ).

We will now use a predefined construction available with MathGraph32 java (Java version 3.3 or later). This construction allows you to create a function curve going through six points specifying the slope of the tangent lines at these six points.

Use menu item *Constructions - Implement predefined construction *. A dialog box pops up. Open the directory named *Curves through points and slopes* and open the construction file named *CurveThroughSixPointsWithTangentSlope*.

A dialog box pops up for the choice of numerical sources objects. These sources objects are the six tangent slopes.

Assign to the six first elements the calculus *m1, m2, m3, m4, m5* and *m6*, and assign to the element number 7 the frame (O,I,J). Validate by **OK**.

You must then (as asked for in the indication text area at the bottom of the window) click on the six points created (in increasing order of abscissa). The curve of the function appears.

Pressing key **F6** that amonf the final objects is a function named *f* and coordinates measures of the clicked points.

Activate the blue color and the thin dotted line style (at the top right of the window).

Create now a point linked to the y-axis (tool ) and name this point *k* (icon ).

Measure the ordinate of point *k* in (O,I,J) frame (tool . This ordinate is represented as yCoord(k,O,I,J).

Create a new calculus named *k* with formula yCoord(k,O,I,J) (tool , use button **Values**).

Create the parallel line to the x-axis through point *k* (icon ).

Let’s now create a function named *g* defined by *g*(*t*)=f(*t*)-k (tool with check box **Draw Curve** not selected).

If necessary, move point *k* to a position where its ordinate is in the range from 0 to 1.

Use now menu item **Calculus -New real calculus - Approximated f(x)=0 root** and fill in the dialog box as show below.

Press **F9** key to reactivate last tool and fill in the dialog box as below :

In the same way press **F9** and fill in the dialog box :

Create now 3 calculus named *ys1, ys2* and *ys3* with formulas *f(s1), f(s2)* and *f(s3)* (tool).

Let’s now visualize the solutions (if existing).

Activate the red color in the color palette.

Create points of coordinates *(s1*; 0), (*s2*,;0) and *(s*3; 0) (icon ).

Let’s now create a display of the value of *s1* linked to point of coordinates (*s1*;0) with tool . Click on the point and fill in the dialog box as shown below :

In the same way create a display of the value of *s2* linked to the point of coordinates (*s2*;0) and a display of *s3* linked to the point of coordinates (*s3*;0).

We want now to calculate and display the number of solutions of the equation *f(x)=k*.

For this we will create four existence tests.

Use menu item**Calculus >> New real calculus >> Test of value existence**. Fill in the dialog box as shown below :

The test *t1* value will be 1 when *s1* exists and 0 otherwise.

In the same way, create two other existence tests of *s2* and *s3*named *t2* and *t3*.

Create now a calculus named N as shown underneath (tool ).

Calculus *N* now contains the number of solutions of the equation f(*x*)=*k*.

Let’s now use an advanced feature of MathGraph32 : dynamic value insertion in a text display.

First click on tool . Click on the top left part of the figure.

Enter the first part of the text as shown underneath :

Click on button **Dynamic value insertion**.

Fill in the new dialog box :

Complete the text field as shown :

On more time click on button **Dynamic value insertion** and fill in the new dialog box :

Complete the text editor and validate:

Now capture point *k* to visualize the solutions of f(x)=k equation and the number of solutions.

Here is the figure animated by MathGraph32 JavaScript motor :

Capture *k* point.