The graph shows g(x), which is a translation of f(x)=x2. Write the function rule for g(x).

Question

Write your answer in the form a(x–h)2+k, where a, h, and k are integers or simplified fractions.
g(x)=

Answer 1

To write the function rule for \( g(x) \) as a translation of \( f(x) = x^2 \), you'll need to determine the values of \( a \), \( h \), and \( k \) based on the transformation observed in the graph compared to the parent function.

1. Identify the vertex of \( f(x) = x^2 \): The vertex is at (0, 0).

2. Find the vertex of \( g(x) \): From the graph (which you have, but I do not), identify the new vertex coordinates, say \( (h, k) \).

3. Determine the value of \( a \):

   • If the graph opens upwards and the vertex has the same shape as \( f(x) \), then \( a = 1 \).
   • If the graph is narrower, \( a > 1 \).
   • If the graph is wider, \( 0 < a < 1 \).
   • If the graph opens downwards, \( a < 0 \).

4. Construct the equation: Insert the values of \( a \), \( h \), and \( k \) into the form \( g(x) = a(x - h)^2 + k \).

For instance, if after inspecting the graph:

• The vertex is moved to \( (2, -3) \) and the parabola opens upwards with no vertical stretch, then you would have \( a = 1, h = 2, k = -3 \). This would give:

\[ g(x) = 1(x - 2)^2 - 3 \]

So the function rule would be:

\[ g(x) = (x - 2)^2 - 3 \]

If you provide the specific translations indicated by the graph, we can arrive at the corresponding specific function.