Question

the graph shows g(x), which is a translation of f(x) = x². write the function rule for g(x).
write your answer in the form a(x - h)² + k, where a, h, and k are integers or simplified fractions.
g(x) =

Explanation:

Step1: Identify the vertex of \( g(x) \)

The vertex form of a parabola is \( a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From the graph, the vertex of \( g(x) \) is at \( (4, 0) \), so \( h = 4 \) and \( k = 0 \).

Step2: Determine the value of \( a \)

The parent function is \( f(x) = x^2 \), which has \( a = 1 \). The graph of \( g(x) \) is a translation of \( f(x) \) with no vertical stretch or compression (since the shape is the same as \( x^2 \)), so \( a = 1 \).

Step3: Write the function rule

Substitute \( a = 1 \), \( h = 4 \), and \( k = 0 \) into the vertex form: \( g(x) = 1(x - 4)^2 + 0 \), which simplifies to \( g(x) = (x - 4)^2 \).

Answer:

\( g(x) = (x - 4)^2 \)