Slope of a line

[Guest]

what are three ways to define the slope of a line?

[adamsmith]

Three ways to define the slope of a line
1.By graph
2.By the formula m=(y1-y2)/(x1-x2)
3.By the equation y=mx+b
where m is the slope of the line.

[Guest]

Depends on the shape of the line

If the line is a straight line then the slope will be the same throughout its length.

By graph....You plot a graph either from data points or an equation of the line....You can measure the slope by the

change in the "Y" direction compared with the corresponding change in "X" direction. Then divide the "Y" dimension

change by the "X" dimension change. The measurement we want is to read the values from the scales of the graph and not

simply measure it with a ruler.

By formula.....You still need a graph or some way of knowing 2 points on the line or the function describing the line.

So you take the "X" and "Y" values of the 2 points and divide the "Y" difference by the "X" difference.....m = (y1 -

y2)/(x1-x2)....= slope of line.

By equation... first you need an equation for the line....If you start with a line (it must be on a graph) and you want

to find the slope of it, you can compare the readings from this line with the standard formula for a straight line

y=mx+b.
You then take 2 readings to get 2 points a short distance apart on the graph, put them in the formula and solve it. If

our readings are, when x=1, y=1, and when x=4, y=3. Then 1 = m.1 + b and 3 = m.4 + b for the 2 points. Subtracting

the 2 equations gives 2 = m.3 so m = 2/3 = 0.667.
So the equation can now be calculated.....3 = 2/3 x 4 + b.....so ...b = 3 - 8/3 = 9/3 - 8/3 = 1/3

So equation of line is Y = 2/3 x + 1/3.


And if the line is curved we are in even a bigger problem.....

Say the graph of Y = "X squared". This is a parabola and its slope is changing as you move around the curve.Its slope

is 2X so it changes as X changes. (as found from calculus, dy/dx)

You could draw a graph of Y = "X squared" by filling a table of data and plotting the points.
And to find the slope at any one point draw a chord (straight line) across the curve and find its slope which would be

an approximate value of the slope at between the points of intersection of the chord and the curve. Then you could let

the chord become smaller and smaller so that in the limit it would tend towards Zero and give you the slope at this

single point. The slope would then in fact be the slope of the tangent to the curve at this point. So measurement

delta"Y" divided by delta"X" would become smaller and smaller until in the limit would become dy/dx of the curve at that

point.

Now back to the original Question.....what are three ways to "define" the slope of a line?

Is this 3 ways of doing the slope or really only one way of doing it. We seemed to need the graph, the formula and the

equation each time....????....and its a bit awkward for curved lines etc
There may be many way to measure and calculate the slope of a line but maybe only one way of defining it.

The slope of a line is the change in "Y" divided by the change in "X"

...........Maybe not so Simple........

[Math Tutor]

Bravo!

[waseem123]

are three ways to define the slope of a line?

[Guest]

read the earlier posts
they are self explanatory

[leesajohnson]

1) By showing it on the graph.
2) By equation.