determine the slope?

Question

Determine the slope of the best-fit line for the following data:

X Y
2 21
4 0
6 -9
8 -12

Answer 1

Go to the website www.mathpad.com and graph it

Answer 2

If you need a best fit straight line, plot these on graph paper, use a string, and visually figure out the line that best matches the plot. Visually figure out what the start and end coordinates for the string are, with your slope being (y2 - y1)/(x2 - x1)

You won't get a good match because the shape of your plot most closely matches the plot of x^2. I would guess the plot hits its minimum around 8, 9, or 10. Your sequence is decreasing by 21, 9, 3 and there's no reason to assume it won't decrease a little more.

You have about 3 realistic plots to choose from, so pick the middle one. You should be able to figure out which way to go if your plot is wrong, so you'll cut your attempts to two. Your minimum value is around -12 instead of 0, so subtract 12.

Plot (x-9)^2 - 12. It looks somewhat promising, even if it's too steep. You can cut the steepness pretty easily. At x=2, your plot is at 21 and the plot of (x-9)^2 is at 37, so 2/3 seems like a reasonable estimate: .67(x-9)^2 - 12

In practice, it cuts too shallow, so I have to move to the left and use (x-8)^2 -12.

That looks a lot more promising, except at 2, I'm still too high (24 instead of 21). Multiplying by 7/8, I get:

.875 * (x-8)^2 - 12

That's very close and might be as close as I'm likely to get.

You differentiate this equation to find your slope. The actual slope will be different at each point of x.

.875 * (x^2 - 16x + 64) - 12 = .875 x^2 -14x + 44

Differentiating, you get:
1.75 x -14

At 2, your slope is -10.5
At 4, your slope is -7
At 6, your slope is -3.5
At 8, your slope is 0

If you know how to chart in Excel, it will find the equation of the line more accurately than my manual method. Just add a 2nd order polynomial trend line and under options, select to display the equation of the trend line. Excel returns the equation:

4.5x^2 - 33.3x +49.5

Edit: Actually, playing with it a little more, 1.1 * (x-7.5)^2 - 12 fits a little better. x-8 is just a bit too shallow of a curve. Once you adjust the curve, you have to adjust the end points to match, as well. That would give a good polynomial of:

1.1x^2 - 16.5x + 49.87

with a derivative of:
2.2x - 16.5

Slope at:

2 -12.1
4 -7.7
6 -3.3
8 +1.1

Answer 3

You first need to plot those points on a graph and draw in a best-fit line. Then, take two points fromthe graph. (It actually doesn't have to pass through ANY of the points given)

You only need to use 2 points on the line to determine the slope. So, pick two and substitute it in the equation:

y2-y1 / x2-x1

Let's use (4,0) and (6,9) (not necessarily points on the best-fit line)

y2-y1 / x2-x1
=(9-0) / (6-4)
=9 / 2
=4.5
The slope (aka gradient) is 4.5

Hope I helped.

Answer 4

slope = -5 1/2 or -11/2

Answer 5

Y2-Y1/X2-X1 OR DIFFERENTIATE

Answer 6

I'd say -2/11ths.