Let the graph of g be a vertical shrink by a factor of 1/3 followed by a translation 5 units to the left of the graph of f(x)=3x^3-x. Write a rule for g.

Answer 1

Let f(x) = 3x^3 - x

Consider an arbitrary constant k and then take k*f(x) and notice that if k > 1 we are shrinking the graph. In particular choosing k = 1/3 we can shrink the graph to (1/3)*f(x) since this means all our "y coordinates" are squashed down to 1/3rd of their previous height relative to the x-axis.

Now suppose h is any fixed positive real number.

If we have any function g(x) then g(x + h) translates our graph h units to the left.

It follows then that g(x - h) would translate our graph h units to the right.

We care about the former case where we want to shift our graph for (1/3)*f(x) 5 units to the left.

Rewrite this function as h(x) = (1/3)*f(x) = x^3 - x/3.

Then the transformation that will shift the graph of h(x) 5 units to the left is given by:

h(x + 5) = (x+5)^3 - (x+5)/3.

Thus finally the function with the graph as described in the problem is given by F(x) = h(x + 5) = (x+5)^3 - (x+5)/3 (where F(x) with a capital F is distinct from the function f with a small f).