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 \((-5, 0)\) (since 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 horizontal shift), \( a = 1 \).

Step3: Substitute \( a \), \( h \), and \( k \) into the vertex form

Substitute \( a = 1 \), \( h = -5 \), and \( k = 0 \) into \( a(x - h)^2 + k \). Remember that \( (x - h) \) becomes \( (x - (-5)) = (x + 5) \) when \( h = -5 \). So the function is \( 1(x + 5)^2 + 0 \), which simplifies to \( (x + 5)^2 \).

Answer:

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