Question

question 14
if $g(x) = x^2$, then which function represents a transformation of the graph of $g(x)$ to the graph shown?
graph of a parabola opening upwards, with vertex at (2, -5), passing through (0, -1) and (4, -1)

Explanation:

Step1: Identify the vertex of the transformed graph

The graph of \( g(x) = x^2 \) has its vertex at \( (0, 0) \). The transformed graph has its vertex at \( (2, -5) \).

Step2: Recall the vertex form of a parabola

The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. For \( g(x) = x^2 \), \( a = 1 \), \( h = 0 \), \( k = 0 \).

Step3: Determine the transformation

Since the vertex is now \( (2, -5) \), we substitute \( h = 2 \) and \( k = -5 \) into the vertex form. So the function becomes \( f(x) = (x - 2)^2 - 5 \). We can expand this to check: \( (x - 2)^2 - 5 = x^2 - 4x + 4 - 5 = x^2 - 4x - 1 \). But the key transformation is the horizontal shift right by 2 units and vertical shift down by 5 units from \( g(x) = x^2 \).

Answer:

The function representing the transformation is \( f(x) = (x - 2)^2 - 5 \) (or expanded form \( f(x) = x^2 - 4x - 1 \))