Question

the graph shows ( g(x) ), which is a translation of ( f(x) = x^2 ). 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) = square )

Explanation:

Step1: Identify the vertex form of a parabola

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 \( f(x) = x^2 \), \( a = 1 \), and its vertex is at \((0, 0)\).

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

From the graph, the vertex of \( g(x) \) is at \((-4, 0)\). So, \( h = -4 \) and \( k = 0 \).

Step3: Determine the value of \( a \)

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

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

Substituting \( a = 1 \), \( h = -4 \), and \( k = 0 \) into \( g(x) = a(x - h)^2 + k \), we get:
\[
g(x) = 1(x - (-4))^2 + 0
\]
Simplifying the expression inside the parentheses:
\[
g(x) = (x + 4)^2 + 0
\]
Which simplifies to:
\[
g(x) = (x + 4)^2
\]

Answer:

\( g(x) = (x + 4)^2 \) (or in the form \( 1(x - (-4))^2 + 0 \), but simplified it is \( (x + 4)^2 \))