Explorations with

the

GEOMETER'S SKETCHPAD

#16

In this exploration you will learn more about a property of lines in a cartesian plane and the standard form equation of a line. Follow these steps to construct a line.

Open a new worksheet and select "Create Axes" and "Snap to Grid" from the **GRAPH** menu. Drag the vertex toward the lower left corner of the screen to give yourself more room in Quadrant I. Now select a point in Quadrant I and name it "adjust line" becuase this will be the point we use to move the line we will create around the graph.

Create a line through this point parallel to the x-axis and another perpendicular to the x-axis, and create the points where these lines intersect the x- and y-axis. (in the sketch below these are points D and E)

Hide the two lines through "adjust line" and use the Line tool to create a line through points D and E. Create a point on this new line and label it P.

Now measure the coordinates of D, E, and P.

Use the Calculator under the MEASURE menu and multiply the x-coordinate of E times the Y coordinate of P then add the product of the y-coordinate of D and the x-coordinate of P. (sounds like a lot but it isn't, we're just multiplying the y-intercept times the x-coordinate of the point and the x-intercept times the y-coordinate of the point and adding them together).

Now move point P along the line and observe what happens.

1) Move the "adjust line" point and see what happens to the value you have calculated.

2) Could you predict the number calculated if you knew the x and y intercepts? Explain how you would do it.

3) What happens to the calculated value if the line is moved closer to (or farther from) the origin? Can you make the slope negative, what happens then?

4) The x-value of P and the y-value of P are the traditional variables we usually call X and Y in the equation of a line. If we put numbers in for the x and y intercepts, the equation might look like this: 5x+3y=15 Can you make this line?

Can you figure out the eqaution of a line that has x-intercept at 3 and y-intercept at 2?