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 parent function \( f(x) = x^2 \) has a vertex at \( (0, 0) \). From the graph of \( g(x) \), the vertex (minimum point) is at \( (4, -2) \).

Step2: Recall the vertex form of a quadratic function

The vertex form of a quadratic function is \( g(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Here, \( h = 4 \) and \( k = -2 \).

Step3: Determine the value of \( a \)

Since \( g(x) \) is a translation of \( f(x) = x^2 \), there is no vertical stretch or compression, so \( a = 1 \).

Step4: Substitute \( a \), \( h \), and \( k \) into the vertex form

Substituting \( a = 1 \), \( h = 4 \), and \( k = -2 \) into \( g(x) = a(x - h)^2 + k \), we get \( g(x) = 1(x - 4)^2 - 2 \), which simplifies to \( g(x) = (x - 4)^2 - 2 \).

Answer:

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