The degree of a polynomial function f(x) is the highest 
power of x found in the expression on the right side of f(x):

f(x) = 5x + 1

is the same as

f(x) = 5x1 + 1

So the highest power of x is 1.  So the degree is 1.

--------------------------------------------------------

y = mx + b where m is the slope and  
           the y-intercept is the point (0,b)

Sometimes y is written as f(x), as in this case.  So in this
and other such cases we have:

f(x) = mx + b where m is the slope and 
           the y-intercept is the point (0,b)

Compare 

f(x) = 5x + 1

to

f(x) = mx + b

and you'll see that m = 5 and b = 1, so the slope is m = 5
and the y-intercept is (0,1).

Edwin