Points on a line: Explain why impossible to get single line eqn thru points

jimsoxfan

New member
Joined
Feb 3, 2017
Messages
1
​​Explain why it is impossible to get a single line function whose plot runs through all of these points.​

(-2.01, -2.59), {-1.74, -2.17}, {-1.5, -1.65}, {-1.27, -1.31}, {-1.04, -1.2}, {-0.84, -1.08}, {-0.45, -0.74}, {-0.35, -0.22}, {0., 0.2}, {0.27, 0.37}, {0.51, 0.45}, {0.73, 0.71}, {0.98, 1.19}, {1.23, 1.68}, {1.58, 1.94}, {1.69, 2.02}, {2.01, 2.2}
 
A good first place to start might be to graph all of the points. What do you see when you do that? Does it immediately appear obvious as to why there's can't be a line between all of the points? Proving this "gut instinct" right and wrong will involve some algebra. You (should) know that the equation of a line is y = mx + b. So, can you find a value of m and b such that it passes through all the points?

Please share with us any and all work you've attempted on this problem, even the parts you know are wrong. Thank you.
 
Here's another approach.

Have you learned the Slope Formula? (Graphically speaking, slope is a measure of "steepness". Numerically speaking, slope is a rate of change -- it measures how fast y is changing, compared to a change in x.)

The Slope Formula uses the coordinates of two points to calculate the slope of the line segment connecting them.

If a set of given points all lie on the same straight line, then the slope of the segments connecting any pair of points must have the same value.

If you find (by using the Slope Formula) that the slope between some points is different than the slope between other points, then that's proof that not all of the points lie on a single straight line. That is, if line segments between points are steeper in some places than others, then the overall line through the points cannot be straight. :cool:
 
​​Explain why it is impossible to get a single line function whose plot runs through all of these points.​

(-2.01, -2.59), {-1.74, -2.17}, {-1.5, -1.65}, {-1.27, -1.31}, {-1.04, -1.2}, {-0.84, -1.08}, {-0.45, -0.74}, {-0.35, -0.22}, {0., 0.2}, {0.27, 0.37}, {0.51, 0.45}, {0.73, 0.71}, {0.98, 1.19}, {1.23, 1.68}, {1.58, 1.94}, {1.69, 2.02}, {2.01, 2.2}
By "line function", do you mean "straight" line? Because it is possible to get a (curved) line through these points. ;)
 
By "line function", do you mean "straight" line? Because it is possible to get a (curved) line through these points. ;)
Mmmmm! Hey Stapel. I've always had an issue with the use of the word "line". I always understood that the mathematical definition of a line is "a set of points which extend infinitely in opposite directions". By this definition, a line is always "straight" (in Euclidean geometry anyway).
 
...I always understood that the mathematical definition of a line is "a set of points which extend infinitely in opposite directions". By this definition, a line is always "straight" (in Euclidean geometry anyway).
Yet, in calculus, one does "line integrals" to find the lengths of non-straight curves.

You're right, though; it would be nice if mathematics were a little more consistent in its terminology. ;)
 
From Wolframalpha:

Euclid defined a line as a "breadthless length," and a straight line as a line that "lies evenly with the points on itself" (Kline 1956, Dunham 1990).

I am not going to pick a fight with Euclid → no-win situation.......
 
By "line function", do you mean "straight" line?
Seems like "single line function" can be single-line function OR single line-function. The former as one slope, and the latter as single equation (i.e., not piecewise). Or the other way around! :cool:
 
Last edited:
Top