When is a function linear and when is it affine?

[tomwilliam2]

Homework Statement

My son (aged 13) received his math test back, and got the following problem wrong:

Which of the following functions is linear?
A: f(x) = -3x + 5
B: g(x) = x^2
C: h(x) = 6x - (1/2)
D: p(x) = 4x/5

The attempt at a solution
He answered B, which is clearly wrong, but it wasn't clear to me why D was supposed to be the right answer. I would have though A and C were also linear. According to his textbook, those are affine functions, and a linear function has to go through the origin. Is this right?
The following website seems to confirm it, but I've always been taught that a linear function is any function where Delta Y / Delta X = constant.
http://www.math.ubc.ca/~cass/courses/m309-03a/a1/olafson/affine_fuctions.htm

Edit: I just found the test and changed the values in it, which doesn't affect the question

 

[StoneTemplePython]

Your post reminds me of this thread (which is worth reading):

https://www.physicsforums.com/threads/linear-relationships.928951/

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For age 13, I would have thought anything but (B) is ok -- all the other answers have lines in them.

 

[tomwilliam2]

Your post reminds me of this thread (which is worth reading):

https://www.physicsforums.com/threads/linear-relationships.928951/

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For age 13, I would have thought anything but (B) is ok -- all the other answers have lines in them.

My thinking exactly, but it does seem that a few of the knowledgeable posters in the thread you indicated agree with my son's textbook. It would appear that the way I was taught (i.e. that f(x) = ax + b is linear) is really a simplification, and they should have called it an affine function. It still strikes me as strange though, given that "linear" should mean you can graphically represent it using a line...

 

[StoneTemplePython]

My thinking exactly, but it does seem that a few of the knowledgeable posters in the thread you indicated agree with my son's textbook. It would appear that the way I was taught (i.e. that f(x) = ax + b is linear) is really a simplification, and they should have called it an affine function. It still strikes me as strange though, given that "linear" should mean you can graphically represent it using a line...

To be crystal clear, what linearity really means is, ##f## is linear iff

##f(a + b) = f(a) + f(b)##
and
##f(\lambda a) = \lambda f(a)##
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graphically that second statement can be interpreted as saying if you double your input, you double your output. This is a simple idea that most students should grasp. (Maybe a simple example like ##y = 7x## vs ##y = 7x + 100## would hit the idea home?)

The first statement can be interpreted as saying that you can draw two line segments on the right (e.g. one line segment is from ##\big(0,0\big) \to \big(a,f(a)\big)## and the other is ##\big(0,0\big) \to \big(b,f(b)\big)##) and add them heads to tails, to get the line segment on the left hand side ##f(a + b)##

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if instead you use a function ##f## where

##f(x) = mx +c ## and ##c \neq 0##

then you break the definition of linearity because
##f(a + b) = m(a+b) + c \neq m(a+b) + 2c = \big(ma + c\big) + \big(mb + c\big) = f(a) + f(b)##

this is why you want the line to go through the origin.
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in your original post you mention ##\frac{\Delta x }{\Delta y}## which is equal to ##= m##. Yes this is key for linearity. But the point is that in linear functions this is the only thing you need to know. With affine functions, you need to juggle intercepts as well.

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These misconceptions apparently bleed all the way into College Algebra.

Personally, I didn't appreciate the difference until Linear Algebra -- where people make distinctions between vector spaces and affine spaces. I'm surprised that they are making the distinction at 13, though it should be helpful later on.

If I was teaching it at that age level, I'd probably say

##f(x) = mx +c ##

gives the equation of a line. Let's plot some examples and see. Now the really good stuff is easiest to use... what's the easiest to use? When the lines go through the origin. (Then give the double input, to double output example.) Mathematicians like easy to use stuff and...

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[tomwilliam2]

To be crystal clear, what linearity really means is, ##f## is linear iff

##f(a + b) = f(a) + f(b)##
and
##f(\lambda a) = \lambda f(a)##
- - - -
graphically that second statement can be interpreted as saying if you double your input, you double your output. This is a simple idea that most students should grasp. (Maybe a simple example like ##y = 7x## vs ##y = 7x + 100## would hit the idea home?)

The first statement can be interpreted as saying that you can draw two line segments on the right (e.g. one line segment is from ##\big(0,0\big) \to \big(a,f(a)\big)## and the other is ##\big(0,0\big) \to \big(b,f(b)\big)##) and add them heads to tails, to get the line segment on the left hand side ##f(a + b)##

- - - -

if instead you use a function ##f## where

##f(x) = mx +c ## and ##c \neq 0##

then you break the definition of linearity because
##f(a + b) = m(a+b) + c \neq m(a+b) + 2c = \big(ma + c\big) + \big(mb + c\big) = f(a) + f(b)##

this is why you want the line to go through the origin.
- - - -
in your original post you mention ##\frac{\Delta x }{\Delta y}## which is equal to ##= m##. Yes this is key for linearity. But the point is that in linear functions this is the only thing you need to know. With affine functions, you need to juggle intercepts as well.

- - - -
These misconceptions apparently bleed all the way into College Algebra.

Personally, I didn't appreciate the difference until Linear Algebra -- where people make distinctions between vector spaces and affine spaces. I'm surprised that they are making the distinction at 13, though it should be helpful later on.

If I was teaching it at that age level, I'd probably say

##f(x) = mx +c ##

gives the equation of a line. Let's plot some examples and see. Now the really good stuff is easiest to use... what's the easiest to use? When the lines go through the origin. (Then give the double input, to double output example.) Mathematicians like easy to use stuff and...

- - - -

Thanks, that's crystal clear.
My son is learning math in the Portuguese system, where they do things a lot more formally than they did where I learned it (the UK). Still, it's good to know this reasoning so I can explain why the correct answer is correct.

 

[mathwonk]

unfortunately the word "linear" is used in both senses. A polynomial such as 2x+7 is linear, because it has no term of degree > 1, and hence in that sense defines a linear function. But in linear algebra, a function is only considered linear if it is also homogeneous, i.e. linear and also has zero constant term; so unfortunately what used to be called homogeneous linear is now usually just called linear, and thia leds to the confusion you experienced. so as usual you always have to ask what is the meaning of the words being used in each situation.