graphing lines

The architect Frank Lloyd Wright used straight lines in many of his buildings.

In fact, he seemed to have a love affair with the straight line. If you examine his work, you will see the use of straight line in almost everything from roofs to foundations to windows to furniture to lamps.

Below are a few pictures from "Fallingwater", a home built near Uniontown Pennsylvania. This home is built overtop of a small waterfall, thus its name, Fallingwater.

This house has a name; Fallingwater, because it is built over top of a waterfall.

This home was built as a summer home for the Kaufmann family of "Kaufmann's" department stores in 1935.

When you are in this home, you can't see the waterfall, but you can HEAR it.

The architect, Frank Lloyd Wright, used a great deal of concrete and native stone to create a dwelling that seemed to "grow" from the surrounding hillside and rocks.

Here is the North-South side elevation on the home 'Fallingwater'. Notice the very strong use of horizontal line.

Extensive use of horizontal line was integrated into the plan. Even the rocks used in the foundation and sidewalls are horizontal in design.

Mr. Wright designed almost everything inside and outside of his houses. Below, we see that he chose to "decorate" with the use of diagonal lines for windows.

a working drawing for a window. Notice the asymetrical, digonal, line use. A window from the 'Robie' house in Chicago

These diagonals added interest to the strong horizontal and vertical lines in the shells of his homes.

NOW MATHEMATICALLY SPEAKING:

It's rather interesting to know that all straight lines can be described with EQUATIONS. They are called linear equations.

Linear equations can always be arranged in the form y = mx + b.
Where m & b are some numbers, and the x & y are unknowns, or variables, as we say in the math world. Other variables can be used instead of x & y, although in math books you see the x and y used frequently.

Here are some examples of linear equations:

y = x (here the m=1, and the b=0)
y = -x (here the m=-1 and the b=0)
y = 2x (here the m=2 and the b=0)
y = -2x + 3 (here the m=(-2) and the b=3)
y = -4x + -9 (here the m=(-4) and the b=(-9))

One way of graphing these linear equations is to set up a table of values.

Here is an example:

Suppose you want to see the line described by y = 2x + 1.

All you need do is select some values along the "x" axis, place them into the equation and find the resulting "y" axis values. Remember, once you have both an "x" and its corresponding "y" value, you have an ordered pair which can be drawn on a graph.

It's convenient to keep track of your ordered pairs for an equation by placing them into a table like so:

x y=2x+1 We choose a value for x in column one, and then figure out the corresponding "y" value and place it in the second column.
Be sure to pick some negative, positive, and zero values for x.
Remember we have only chosen 5 points on this line here. The line has a lot more points, actually an infinite number of points.
The beauty is that when making a chart, you may select any values for x that you like. So always select numbers that are easy to work with.

-3 2(-3)+1=-5

-1 2(-1)+1=-1

0 2(0)+1=1

1 2(1)+1=3

3 2(3)+1=7

Then we graph these values, and it will look like this:

y=2x+1.gif (7464 bytes)

Click HERE for some practice on graphing.