Question

suppose the graph of ( f ) is given. describe how the graph of each function can be obtained from the graph of ( f ).
(a) ( y = f(x + 5)+2 )
shift 2 units to the right and 5 units upward
shift 2 units to the right and 5 units downward
shift 5 units to the left and 2 units downward
shift 5 units to the right and 2 units upward
shift 5 units to the left and 2 units upward
(b) ( y = f(x - 4)-3 )
shift 4 units to the right and 3 units downward
shift 3 units to the left and 4 units upward
shift 3 units to the right and 4 units downward
shift 4 units to the right and 3 units upward
shift 4 units to the left and 3 units downward

Explanation:

For part (a), when we have a function \(y = f(x + 5)+2\), according to the rules of function transformations:

• For the horizontal transformation, if we have \(y = f(x + h)\) (\(h>0\)), the graph of \(y = f(x)\) is shifted \(h\) units to the left. Here \(h = 5\), so there is a shift of 5 units to the left.
• For the vertical transformation, if we have \(y=f(x)+k\) (\(k > 0\)), the graph of \(y = f(x)\) is shifted \(k\) units upward. Here \(k = 2\), so there is a shift of 2 units upward.

For part (b), when we have a function \(y = f(x - 4)-3\), according to the rules of function transformations:

• For the horizontal transformation, if we have \(y = f(x - h)\) (\(h>0\)), the graph of \(y = f(x)\) is shifted \(h\) units to the right. Here \(h = 4\), so there is a shift of 4 units to the right.
• For the vertical transformation, if we have \(y=f(x)+k\) (\(k<0\)), the graph of \(y = f(x)\) is shifted \(|k|\) units downward. Here \(k=- 3\), so there is a shift of 3 units downward.

Answer:

(a) shift 5 units to the left and 2 units upward
(b) shift 4 units to the right and 3 units downward