Question

the graph below shows the function $g(x) = x^2$ and a transformed function. which of the following accurately describes the transformation?

a. the function $g(x) = x^2$ is shifted 3 units to the right and 5 units up to obtain the function $f(x) = (x - 3)^2 + 5$

b. the function $g(x) = x^2$ is shifted 3 units to the right and 1 unit up to obtain the function $f(x) = (x - 3)^2 + 1$

c. the function $g(x) = x^2$ is shifted 3 units to the right and 1 unit up to obtain the function $f(x) = (x + 1)^2 + 3$

d. the function $g(x) = x^2$ is shifted 3 units to the right and 1 unit up to obtain the function $f(x) = (x + 3)^2 + 1$

Explanation:

Step1: Recall Vertex Form of Parabola

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

Step2: Identify Vertex of Transformed Function

From the graph, the vertex of the transformed function is at \((3, 1)\). So \( h = 3 \) and \( k = 1 \).

Step3: Determine Transformation and Equation

Shifting \( g(x) = x^2 \) 3 units right (since \( h = 3 \), \( x \to x - 3 \)) and 1 unit up (since \( k = 1 \)) gives \( f(x) = (x - 3)^2 + 1 \), which matches option B.

Answer:

B. The function \( g(x) = x^2 \) is shifted 3 units to the right and 1 unit up to obtain the function \( f(x) = (x - 3)^2 + 1 \)