SAT function question?

Test Preparation SAT Preparation
1

The figures above show the graphs of the function f and g. The function f is defined by f(x)=X^3-4x. The function g is defined by g(x)= f(x+h)+k, where h and k are constants. What is the value of hk?
Graph of f(x) has points (1,3) and (1,-3) which are changed to (2,1) and (4,-5) on graph of g picture.

So. first, I found what f(x+h) is–> (x+h)^3 - 4(x+h) --> (x+h)^3 -4x -4h
Now I plug that into the g(x) function for f(x+h) ----> g(x)= (x+h)^3 -4x -4h + K
Alright, and I know the graph of f to g shifts 3 units to the right (h=-3) and 2 units down (k=-2).

So…I plug the h in… g(x)= (x-3)^3 -4x + 12 + K …notice it’s [12+K] , if the graph shifts only 2 down then it should have a constant of -2 …so if I set 12+k=-2, I get k= -14.

And -14 x -3 is obviously not 6 (which is the answer). The thing is, I know how they came to the answer of six, collegeboard basically said h was -3 and k was -2 …BUT if K is -2 …then the function of g(x) would actually = (x-3)^3 -4x +10 …and obviously the graph doesn’t shift up 10…SO WHAT’S UP WITH THIS. what am I missing?

2

Ok …that’s weird…I just graphed " g(x)= (x-3)^3 -4x +10" into the calculator and got the same graph from the picture…so that means, yes, k does =-2 because 12-2=10 …but I still have a question: how does a +10 to the equation translate into f(x) shifts two downward…?? Shouldn’t the function be (x-3)^3 -4x -2 for the graph to shift two down?

3

f(x) has points (1,-3) and (-1,3) , not ( 1,3)

4

if you use the figure you are given , you should be able to answer to the question within 15 sec. f(x)+1 means that it goes upward 1 unit. f(x-1) means that it goes to the right 1 unit, so if you see the figure , you recognize that the function g(x) goes down 2 unit and shifts x coordinate right in a positive direction. Therefore, the function must be g(x)=f(x-3)-2, so K= -2 , h=-3. -3 X -2 = 6.
Correct answer is “6”