Mathematics
What do all linear functions of the form f(x) = ax + a, a > 0, have in common?
Polynomials
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Answer
The given linear function is f(x) = ax + a, where a > 0.
This can be rewritten as f(x) = a(x + 1).
To find a point common to all such functions, we look for a value of x that makes f(x) the same for any value of a.
When x = -1:
f(-1) = a(-1) + a = -a + a = 0
So, for every value of a > 0, the line f(x) = ax + a passes through the point (-1, 0).
Also, the slope of each line is 'a' (positive, since a > 0), so all these lines rise from bottom-left to top-right and represent linear growth.
The y-intercept of each line is 'a', which is the same as the slope.
Hence, all linear functions of the form f(x) = ax + a, where a > 0, pass through the common point (-1, 0) on the x-axis. They all have positive slopes (so they all represent linear growth), and the slope of each line is equal to its y-intercept.
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