Description

which translation maps the graph of the function f(x) = x2 onto the function g(x) = x2 + 2x + 6 ?

The translation that maps the graph of the function ( f(x) = x^2 ) onto the function ( g(x) = x^2 + 2x + 6 ) involves shifting the graph of ( f(x) ) one unit to the left and five units upward.

**Step-by-Step Explanation:**

1. **Convert ( g(x) ) to Vertex Form:**
– Start with ( g(x) = x^2 + 2x + 6 ).
– Complete the square:
[
g(x) = (x^2 + 2x) + 6
]
[
= (x^2 + 2x + 1) – 1 + 6
]
[
= (x + 1)^2 + 5
]
– Thus, ( g(x) = (x + 1)^2 + 5 ).

2. **Identify the Vertex:**
– The vertex form of ( g(x) ) is ( (x + 1)^2 + 5 ), which corresponds to ( (x – (-1))^2 + 5 ).
– Therefore, the vertex of ( g(x) ) is at ( (-1, 5) ).

3. **Determine the Translation:**
– The original function ( f(x) = x^2 ) has its vertex at ( (0, 0) ).
– To move the vertex from ( (0, 0) ) to ( (-1, 5) ), the graph must be shifted 1 unit to the left (since ( h = -1 )) and 5 units upward (since ( k = 5 )).

**Conclusion:**
The graph of ( f(x) = x^2 ) is translated 1 unit to the left and 5 units up to obtain the graph of ( g(x) = x^2 + 2x + 6 ).