Question

Do both x and y values have to have a constant rate of change to be a linear function?

Answer 1

define a linear function, it has different meanings in different contexts

Answer 2

a linear function is either a line equation ... or a function such that f(a+b) = f(a)+f(b), and f(kx) = k f(x) for some constant k

Answer 3

@PeachRings So they both have to have constant rate of change right?

Answer 4

@amistre64 I don't know what you're talking about. lol I mean for linear functions in tables.

Answer 5

a linear function has a common different, a constant slope if we are looking at table values

Answer 6

so yeah, x and y would have a constant rate of change.

Answer 7

0 is still a rate of change :)

Answer 8

But for y values, they have to all have the same amount of change right?

Answer 9

i know how im reading 'amount of change' but im not sure how your reading it.

lets say y=1,3,4,7,12 this of course does not appear like a constant rate of change, but that all depends on the x values that relate to them.

Answer 10

if given a table, its points form a line if the slope between all the stated points is the same.

Answer 11

So if the x and y values are data and the y values are not consistent in amounts but they are with a x value it's linear. Sorry for all the questions.

Answer 12

@amistre64

Answer 13

do you have a table to work from? or is this just a mental practice?

Answer 14

lets assume we have a line y=mx+b and see if we cant get a y/x = setup

y-b = mx

(y-b)/x = m is constant ... if b does not equal zero then we can prolly assess it from table values

Answer 15

\[\frac{y_o-b}{x_o}=\frac{y_1-b}{x_1}\]
\[\frac{y_o}{x_o}-\frac{b}{x_o}=\frac{y_1}{x_1}-\frac{b}{x_1}\]
\[\frac{y_o}{x_o}-\frac{y_1}{x_1}=\frac{b}{x_o}-\frac{b}{x_1}\]
\[\frac{y_o}{x_o}-\frac{y_1}{x_1}=b\left(\frac{1}{x_o}-\frac{1}{x_1}\right)\]
\[\frac{y_o~x_1-y_1x_o}{x_o~x_1}=b\left(\frac{x_1-x_o}{x_o~x_1}\right)\]
\[\frac{y_o~x_1-y_1x_o}{x_1-x_o}=b\]
so, if given a set of points, then we have some work to assess if they form a line if this condition holds for some random points on a line

Answer 16

I got it now. Thanks for all your help and great effort :)

Answer 17

youre welcome