The graph of f(x) = x2 is translated to form g(x) = (x – 5)2 + 1.

Question

On a coordinate plane, a parabola, labeled f of x, opens up. It goes through (negative 2, 4), has a vertex at (0, 0), and goes through (2, 4).

Which graph represents g(x)?

On a coordinate plane, a parabola opens up. It goes through (2, 10), has a vertex at (5, 1), and goes through (8, 10).
On a coordinate plane, a parabola opens up. It goes through (2, 8), has a vertex at (5, negative 11), and goes through (8, 8).
On a coordinate plane, a parabola opens up. It goes through (negative 8, 10), has a vertex at (negative 5, 1), and goes through (negative 2, 10).
On a coordinate plane, a parabola opens up. It goes through (negative 8, 8), has a vertex at (negative 5, negative 11), and goes through (negative 2, 8).
Mark this and return

Answer 1

To determine which graph represents \( g(x) = (x - 5)^2 + 1 \), we need to understand the transformations applied to the original function \( f(x) = x^2 \).

1. Vertex Translation:

   • The original parabola \( f(x) = x^2 \) has a vertex at (0, 0).
   • The transformation \( g(x) = (x - 5)^2 + 1 \) means the parabola is shifted to the right by 5 units and up by 1 unit.
   • Consequently, the vertex of \( g(x) \) is at \( (5, 1) \).

2. Direction of Opening:

   • Since the leading coefficient of \( (x - 5)^2 \) is positive, the parabola opens upward.

3. Points on the Graph:

   • To find additional points on the graph of \( g(x) \), we can calculate:
     • When \( x = 5 \), \( g(5) = (5 - 5)^2 + 1 = 1 \) (the vertex).
     • When \( x = 4 \), \( g(4) = (4 - 5)^2 + 1 = 2 \).
     • When \( x = 6 \), \( g(6) = (6 - 5)^2 + 1 = 2 \).
     • When \( x = 3 \), \( g(3) = (3 - 5)^2 + 1 = 5 \).
     • When \( x = 7 \), \( g(7) = (7 - 5)^2 + 1 = 5 \).

Based on this information, we now check the options provided:

1. First Option:

   • Vertex at \( (5, 1) \), goes through \( (2, 10) \) and \( (8, 10) \). This option seems to match as it has the vertex at (5, 1).

2. Second Option:

   • Vertex at \( (5, -11) \). Does not match.

3. Third Option:

   • Vertex at \( (-5, 1) \). Does not match.

4. Fourth Option:

   • Vertex at \( (-5, -11) \). Does not match.

Conclusion: The graph representing \( g(x) \) is the first option, which has a vertex at \( (5, 1) \) and opens upward, consistent with the transformations applied to \( f(x) \).