Question

the graph shows g(x), which is a translation of f(x) = x². write the function rule 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 form

The vertex form of a parabola is \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. For the parent function \( f(x)=x^2 \), the vertex is \((0, 0)\).

Step2: Find the vertex of \( g(x) \)

From the graph, we can see that the vertex of \( g(x) \) is at \((-3, 5)\). So \( h=-3 \) and \( k = 5 \).

Step3: Determine the value of \( a \)

Since \( g(x) \) is a translation of \( f(x)=x^2 \) (no vertical stretch or compression, just translation), the value of \( a \) is the same as in \( f(x) \), which is \( 1 \).

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

Substitute \( a = 1 \), \( h=-3 \), and \( k = 5 \) into \( g(x)=a(x - h)^2 + k \). We get \( g(x)=1(x - (-3))^2+5=(x + 3)^2+5 \).

Answer:

\( g(x)=(x + 3)^2+5 \)