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.

Explanation:

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

The vertex form of a parabola is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex. From the graph, the vertex of \( g(x) \) is at \((5, 4)\) (since the minimum point of the parabola is at \( x = 5 \), \( y = 4 \)).

Step2: Determine the value of \( a \)

The parent function is \( f(x) = x^2 \), which has \( a = 1 \). Since there is no vertical stretch or compression (the parabola has the same width as \( f(x) = x^2 \)), \( a = 1 \).

Step3: Write the function rule

Substitute \( a = 1 \), \( h = 5 \), and \( k = 4 \) into the vertex form \( y = a(x - h)^2 + k \). So we get \( g(x) = 1(x - 5)^2 + 4 \), which simplifies to \( g(x) = (x - 5)^2 + 4 \).

Answer:

\( g(x) = (x - 5)^2 + 4 \)