find a formula for ...

[mathdaughter]

I know how to do the question a, but don't know how to do the b.

This is not an assignment. I am trying to understand what my daughter is learning. Anybody can help me? Thanks

 

[Steven G]

What does the function f(x) do to what is in the parenthesis (which is x). Answer: It multiplies it by a and then adds b.
We do the same for f(x+1), we multiply x+1 by a and then add b. Hence f(x+1) = a(x+1) + b. Now subtract f(x) from that.
Show us what you get.

 

[mathdaughter]

are you saying the formula should be like this:

f(x + 1) - f(x)
=(a(x+1) + b) - (ax + b)
= ax + a + b -ax -b
= a

which is the delta column in the question a table? but it does not look like a formula, right? then I will have to make a recursive definition based on above result?

 

[lev888]

are you saying the formula should be like this:

f(x + 1) - f(x)
=(a(x+1) + b) - (ax + b)
= ax + a + b -ax -b
= a

which is the delta column in the question a table? but it does not look like a formula, right? then I will have to make a recursive definition based on above result?

Draw a graph of any ax+b function and note the difference between f(1) and f(2), f(2) and f(3), etc. What do you think?

 

[mathdaughter]

I think I know the graph part. f(x) will be a line. f(x + 1) will be another line and they are parallel. I don't know the formula part. what the formula should be?

 

[lev888]

I think I know the graph part. f(x) will be a line. f(x + 1) will be another line and they are parallel. I don't know the formula part. what the formula should be?

It's one line. f(x) is y value at x. f(x+1) is y value at x+1. So look at the line and note how y value changes as x is increased by 1.

 

[Steven G]

are you saying the formula should be like this:

f(x + 1) - f(x)
=(a(x+1) + b) - (ax + b)
= ax + a + b -ax -b
= a

which is the delta column in the question a table? but it does not look like a formula, right? then I will have to make a recursive definition based on above result?

It might not look like the correct formula but the algebra does not lie.
Check to see what f(4)-f(3), f(7)-f(6), f(3.2), f(3.2) - f(2.2) all equal. You can use the formula directly or you can use the graph, as already pointed out.
Note that in f(anything) - f(anything-possibly a different anything) that the constant b will ALWAYS cancel out. In the end in computing f(x+1) - f(x) you can actually use f(x) = ax (again use can use f(x) = ax ONLY when computing f( one thing) - f(another thing) )
Now maybe it is clear that f(x+1) - f(x) = a.

For example f(7) - f(2) = (7a-b) - (2a-b) = 7a-2a. If we defined g(x) = ax, then g(7) - g(2) = (7a) - (2a). Note that we got the same answer and that the answer was simply 7-2 times a. In f(x+1)-f(x) the answer will be ((x+1)-(x))*a = (1)a = a

 

[Steven G]

A deeper understanding of this problem comes from considering the definition of slope and the slope of the given line (ax+b is the equation of a line)

The slope of this line is a (it comes from the slope being m in the formula y=mx+b)

Now m = a = change in f(x)/ change in x = [f(x+1) - f(x)]/[(x+1)-(x)] = f(x+1) - f(x)]/[1] = f(x+1) - f(x)]. That is f(x+1) - f(x)] = a !!