In this tutorial, we are going to use Geometer’s Sketchpad to explore the graph of the function **y = mx + b** where** m** and** b **are real numbers. First, we are going to type each equation manually, but later, we are going to use sliders to see the relationship between the parameters **m** and **b **and the appearance of the graph.

**Steps Graphing Equations**

- Open
**Geometer’s Sketchpad**. - Click the
**Graph**menu from the menu bar and click**Define Coordinate System**from the list. - To graph the function
**y = 2x**, click the**Graph**menu, then click**Plot New Function**to display the*New Function**dialog box*. - In the
*New Function**dialog box*, type**2x**, then click the**OK**button.

Using steps **1** through **4**, graph the following functions and observe how the value of **m** affects the graph of **y = mx**

- y = 3x
- y = 4x
- y = 5x
- y = 10 x
- y = – 2x
- y = – 4x
- y = -6x
- y = – 10x

Graph the following functions and observe how **b** affects the graph of the function **y = mx + b.**

- y = 2x + 3
- y = 2x + 1
- y = 2x + 5
- y = 2x – 1
- y = 2x – 4
- f. y = 2x – 10

**Creating a Slider**

There is a better way to explore the relationship of the parameters of functions and their graphs. Instead of typing each equation, we can use the sliders to assign values to parameters like **m **and **b. **A slider is a visual representation of a number. For instance, if you have a slider **m** with domain **-10** through **10**, then moving the slider rightward will increase the value of **m**. The slider that we will create here is very similar to Graphs and Sliders 1 and Graphs and Sliders 2 posts in the GeoGebra Tutorial Series.

The construction of slider in Geometer’s Sketchpad is somewhat different compared to the slider in GeoGebra. We will use the idea of ratio in creating a slider here. To create a slider, we will construct segment ** AB**, and construct point

*on*

**C****. We will divide the measure of**

*AB***by the measure of**

*AC***,then multiply it to**

*AB***20**. This means that our minimum value is

**0**and our maximum number is

**20**. To facilitate negative values, we will subtract

**10**from result of our computation. This means that our minimum value is

**0 – 10 = -10**and our maximum value is

**20 – 10 = 10.**

If you want to extend the domain of your slider, you just multiply the quotient of ** AC** and

**by your desired number and subtract half of that desired number from the product.**

*AB***Steps in Constructing a Slider**

- Open
**Geometer’s Sketchpad.** - To show the coordinate axes, click the
**Graph**menu from the menu bar and click**Define Coordinate System**from the drop-down list. - To construct our slider, click the
*Segment tool*from the toolbox, and construct a horizontal segment on the drawing area. - To display the label of the two points, select the two points, click the
**Display**menu from the menu bar and click**Show Labels**from the list. - To construct point
on*C*, click the*AB***Point tool**and click segment(not the points).*AB* - Display the label of point
.*C,*by right clicking it and choosing**Show lab****el** - For our computation of the value of
**m**, we first measure the value ofand*AC*. To measure*AB*, select points*AC*and*A*(be sure that only the two points are selected), click the*C***Measure**menu and click**Distance**from the list. - To measure
, select points*AB*and*A*, then click the*B***Measure**menu and click**Distance**from the list. - To find the value of
**m**, we divideby*AC*, multiply the result to*AB***20**and the subtract**10**. To do this, click the**Measure**menu and click**Calculate**from the list. - Click the text on the drawing area displaying the measure of
, click the*AC***÷**button from the*New Calculation*dialog box, click the label displaying the measure of, click*AB******from the*New Calculation*dialog box, then type**20-10**, then click the**OK**button on the dialog box when finished. This will be our value of.*m* - Move point
and observe what happens to the value of**C**. If you want to Edit your calculation, just click the*m*tool, right click the the value of m, then click*Arrow***Edit Calculation**. - To graph
**y = mx**, click the**Graph**menu, then click**Plot New Function**to display the*Plot New Function dialog box*. - While the
*Plot New Function*box is displayed, click the label containing the value of, click the * button, click the*m**x*(or type*x*), then click the**OK**button. If you have followed the steps correcty, the graph of**y = mx**should appear in your coordinate system. - Hide points
and*A*and the labels containing the values of*B*and*AB*by clicking the*AC***Display**menu and click**Hide Objects.** - Move point
. What do you observe? What relationship can you conclude between the value of*C*and the appearance of the graph of the function?*m*

**Exercise:**

- Construct another slider for the value of
**b**. - Construct a graph that will display the value of
**f(x) = mx + b**. - Describe the effect of
**b**in the graph of the function**f(x) = mx + b**.