Calculus, Algebra

Question

What kind of transformation converts the graph of 

f

(

x

)

=

3

x

–

9

 into the graph of 

g

(

x

)

=

x

–

3

?

What kind of transformation converts the graph of f(x)=3x–9 into the graph of g(x)=x–3?

Answer 1

The first thing I notice about f(x) = 3x - 9 is that the two constants (namely 3 and 9) are both multiples of 3. That is, if we factor 3 from both of our terms we get:

f(x) = 3(x - 3).

Now, as g(x) = x - 3, we see that in order to obtain this answer, we must divide our original function by the 3 we have factored out. As such, to obtain g(x) from f(x) the most appropriate transformation would be a vertical shrink by a factor of 1/3.

Or, g(x) = (1/3)f(x).

Answer 2

The difference is the straight line y(x)=3x-9 shifts 6 units up along the y axis from -9 to -3 along the y axis, and in the same time its slope gets flatter by a factor of 3 such that the slope of y = 3x-9 is 3 and the slope of

g(x) = x-3 is 1