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 quadratic function is \( a(x - h)^2 + k \), where \( (h, k) \) is the vertex. From the graph, the vertex of \( g(x) \) is at \( (5, 0) \) (since the parabola opens upwards and the minimum point is at \( x = 5 \), \( y = 0 \)).

Step2: Determine the value of \( a \)

The parent function is \( f(x) = x^2 \), which has \( a = 1 \). Since \( g(x) \) is a translation (no vertical stretch or compression, just a horizontal shift), \( a = 1 \).

Step3: Write the function rule

Substitute \( a = 1 \), \( h = 5 \), and \( k = 0 \) into the vertex form: \( g(x) = 1(x - 5)^2 + 0 \), which simplifies to \( g(x) = (x - 5)^2 \).

Answer:

\( g(x) = (x - 5)^2 \)