Question

algebra 1 > y4 transformations 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 parent function \( f(x) = x^2 \) has a vertex at \( (0, 0) \). For \( g(x) \), looking at the graph, the vertex (the minimum point of the parabola) is at \( (-3, 8) \). In the vertex form \( a(x - h)^2 + k \), the vertex is \( (h, k) \), so \( h = -3 \) and \( k = 8 \).

Step2: Determine the value of \( a \)

Since the parabola \( g(x) \) is a translation of \( f(x) = x^2 \) (no vertical stretch or compression, just a translation), the value of \( a \) remains \( 1 \) (because the shape is the same as \( f(x) \), so the coefficient of \( (x - h)^2 \) is \( 1 \)).

Step3: Write the function rule

Substitute \( a = 1 \), \( h = -3 \), and \( k = 8 \) into the vertex form \( a(x - h)^2 + k \).

$$ g(x) = 1(x - (-3))^2 + 8 = (x + 3)^2 + 8 $$

Answer:

\( (x + 3)^2 + 8 \)