(a) The graph of f(x) is given, it is required to translate this graph to get the graph of y=f(x+4).
The given graph of f(x) is shown below:
Recall that the graph of y=f(x-k) is a horizontal translation of the graph of y=f(x), where for k>0, the graph is shifted right and for k<0, the graph is shifted left.
Rewrite y=f(x+4) as follows:
[tex]\begin{gathered} y=f(x+4)_{} \\ \Rightarrow y=f(x-(-4)) \end{gathered}[/tex]
Notice that the value of k here is -4 which is negative (less than zero).
This implies that to get the graph of y=f(x+4), the graph of y=f(x) has to be shifted left by 4 units.
Shift the given graph of y=f(x) to the left by 4 units to get the required graph of y=f(x+4) as follows:
(The graph of y=f(x+4) is shown in blue).
(b) The graph of g(x) is given, it is required to translate this graph to get the graph of y=g(x)+3.
The given graph of g(x) is shown below:
Recall that the graph of y=f(x)+k is a vertical translation of the graph of y=f(x), where for k>0, the graph is shifted up and for k<0, the graph is shifted down.
The graph of y=g(x)+3 is required.
Notice that here, k=3 which is positive (greater than zero). It implies that the graph of g(x) has to be shifted up by 3 units.
Shift the given graph of y=g(x) up by 3 units to get the required graph of y=g(x)+3 as shown below:
(The graph of y=g(x)+3 is shown in purple).
