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If f(x) is linearly related to x and f(x+4)-f(x)=32, what is the slopes of the graph of f?

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3 Answers

First you need to know that to find the slope the equations must be linearly related. This is stated in your question. 

Slope is found through determining the change in f(x) and dividing by the change in x.

This can be written as:  m= f(x)2-f(x)1
                                           x2-x1

So looking at your problem: f(x+4)-f(x)=32

We know that: x2=x+4 and x1=x

We are then able to substitute our knowns to solve for slope m.

m= f(x)2-f(x)1f(x+4)-f(x) =  _ 32 _ =    32   32 = 8
          x2-x1          (x+4)-(x)      x+4-x     x-x+4     4

Thus your slope, m=8.

 You can rewrite any similar f(x) equations that need to have the slope solved into the above format and solve for m. By breaking the problem down into its smaller parts you can tackle and solve any math problem.

A slope is a measurement of a change in f(x) over the change in x. f(x+x) - f(x) = 32 tells you two things. First it tells you that the change in x is 4, which is pretty straight forward. The difference between x and x+4 is 4. Secondly, it tells you that the change in f(x) between x and x+4 is 32. So, change in f(x) over change in x is 32/4 or 8, which is the slope of f(x).

Comments

Nice sweet thinking man's answer, Patrick,with no equations needed or really manipulations. [slope = rise/run is the concept]  

Where did you get your math degree; there in Rochester Tech?

Comment

Since ƒ is linearly related to  x   then it means that f is of the form  f(x) = m x + b  where m is the slope of the graph of  f  and  (0, b) is the y-intercept of the graph of  f.

Now,  f(x+4) = m (x + 4) + b = mx + b + 4m   and   f(x + 4) - f(x)  = (m x + b) - (mx + b) + 4m = 4m.

Since  we are told that f(x + 4) - f(x) = 32, then it means that 4m = 32 and so  m = 8.  Hence the slope of the graph of f is  8.