*Determining the sign of a linear function*

*Determining the variations of a linear function.*

*Justifying sign from variations*

*Determining the sign of a product*

We take the example of the sign of the product of two linear expressions.

Two ways for determining the sign of a linear expression are demonstrated:

- one uses results about linear expressions' sign (method 1),
- the other uses results about variation of linear functions (method 2).

Then we will use the rule of signs of a product or of a quotient for the signs of *f*.

### Finding sub-expressions

Create the function *f* . You can use** ****function by formula**. However, it is important to check the domain in order that the mini-diagram appear. In the *expressions list*, click on the expression of *f*.

Figure 62 - Menu Calculate / Sub-expression

Click on the **Calculate** button, in the contextual menu which appears, choose **Sub-expressions**. The following *box* appears, it lists the sub-expressions.

Figure 63 - Sub-expressions list

The selection of list's elements is as in Windows™ explorer :

Keeping « Ctrl » key pressed, click on non-consecutive elements in order to select them.

Keeping « Shift » or key pressed, click on first, then on last element of the list to be selected; the **All** button selects all the elements.

In this example choose** All**, then click **OK**. The *expressions* and *functions lists* are completed by two functions named *f** _{0}* and

*f*

*defined from the selected expressions.*

_{1}

Figure 64 - Sub-expressions (functions list)

Figure 65 - Sub-expressions (expressions list)

### Determining the sign of a linear function

Highlight *f** _{1}* in the

*functions list*as seen above.

In the **Justify** menu, choose **Sign: linear**.

Figure 66 - Menu Justify / Sign - linear

A *dialog box* appears.

Figure 67 - Box "Sign - linear - Condition of application"

Fill in *Coeff of x* and *null at *then click on the **Evaluate** button. If there is an error, information appears in the *Notepad*.

Figure 68 - Sign linear completed box with error

Figure 69 - Box with correct answer

If the correct values are entered, after clicking **Evaluate**, a new *box* opens.

Figure 70 - Conclusion box for the sign of linear function

On each line, successive clicks in the blank box make *negative* or *positive* appear; evaluate your propositions by clicking the **Evaluate Prop** button.

Figure 71 - Conclusion box with first line filled

When propositions are correct, the **OK** button is active, please press it.

Figure 72 - Conclusion box with correct answers

The zero of *f** _{1}* is added in the

*x-values list*and in the

*functions list*the sign of

*f*

*is showed with green bullets (positive if the bullet is above the horizontal line passing by 0, else negative).*

_{1}

Figure 73 - Displaying sign of *f _{1}*

### Determining the variations of a linear function

In order to determine the sign of (π – 2x), create the zero of the expression *f** _{0}* (π/2) in the

*x-value list*(click New Value) then select

*f*

*in the*

_{0}*functions list*.

In the **Justify** menu, choose **Variations: reference functions**.

Figure 74 - Menu Justify / Variations - reference functions

This *box* opens:

Figure 75 - Box "Variations - reference functions - Conditions of application"

You can choose the type of function in the list, and you will get a graphical representation of the chosen function’s type. Click on **Evaluate**.

After a correct answer, a *Conclusion box* opens.

Figure 76 - Conclusion box for variations

For each line, click on the blank box in order to make *decreasing* or *increasing* appear. Evaluate your propositions with the **Evaluate Prop** button. When propositions are correct, the **OK** button is active, please press it.

Figure 77 - Conclusion box for variations with correct answers.

In the *functions list*, results appear in the mini-diagrams (arrows).

Figure 78 - Displaying variations of *f _{0}*

### Justifying sign from variations

After variations, it is needed to justify the sign; in the **Justify** menu, choose **Sign: known variations**.

Figure 79 - Menu Justify / Sign - known variations

The *box Sign: Known variations: application conditions* opens. At one of the values in the drop down menu, indicate if the function is *negative*, *zero* or *positive*. Here, choose *x2* and **zero function**. It means that you recognised that the function is decreasing and null at *x2, *and then Casyopée will know that the function is positive for x< x2 and negative for x> x2. When the proposition is correct, the **Evaluate** button is active, please press it.

Figure 80 - Box "Sign - known variations - Conditions of application"

A *Conclusion box* opens. For each line, click on the blank box in order to make *negtive* or *positive* appear. Evaluate your propositions with the **Evaluate Prop** button. When propositions are correct, the **OK** button is active, please press it.

Figure 81 - Conclusion box for sign of a function with known variations

The signs of *f** _{0}* are denoted by the position of arrows. Above the line passing by 0, the function is positive and under, the function is negative.

Figure 82 – Sign diagram of *f _{0}*

### Determining the sign of a product

Finally, in the *functions list*, select *f*, and then click on the **Justify** menu, choose **Sign: product, quotient**.

Figure 83 - Menu Justify / Sign - product, quotient

At the bottom of the *dialogue box*, select the two lines, because *f* is the product of *f** _{0}* and

*f*

*.*

_{1}

Figure 84 - Box "Sign - product, quotient - Conditions of application"

Click on **Evaluate**, the *Conclusion box* opens. The meaning of this window has already been explicated. After pressing **Evaluate Prop**. and **OK**, the results appear as signs in the *functions list*.

Figure 85 - Conclusion box for the sign of a product

In the *functions list*, the table of sign is totally filled. The *Notepad* allows to keep track of the different steps and to write a justification.

Figure 86 - Sign table of functions