Graph Transformation

Question

The graph of f(x) contains the point (−1,6). What point must be on each of the following transformed graphs? Enter points as (a,b)(a,b) including the parentheses.

(a) The graph of f(x−7) must contain the point 

(b) The graph of f(x)−5 must contain the point

(c) The graph of f(x+6)+3 must contain the point 

Answer 1

I agree with Raymond's response. I just wanted to add something, because the horizontal shifts often seem backwards to some, such as (a) which has (x - 7) but that shifts to the right.

The example given says f(x) = (-1,6). So, the function f gives an output of 6 when the input is -1.

In (a) we're asked to consider f(x-7). What x-value do I need here to make the input equal to -1? Clearly x needs to equal 6. So, to get the same output, the input for f(x-7) has to be 6, which is 7 larger than the input for f(x). That's why the graph gets shifted 7 to the right.

And, similarly, as seen in (c), f(x+6) is a shift to the left of 6.

Answer 2

f(x) contains (-1,6)

then f(x-7) is f(x) shifted horizontally to the right by 7

so f(x-7) contains the point (-1+7, 6) = (6,6)

f(x)-5 is a shift vertically down 5, and contains the point (-1,6-5) = (-1,1)

f(x+6)+3 is a shift left 6, and up by 3, containing the point (-1-6, 6+3) = (-7,9)