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

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 Transformed Vertex

From the graph, the transformed parabola has a vertex at \((3, 1)\). So \( h = 3 \), \( k = 1 \).

Step3: Analyze Horizontal Shift

A horizontal shift of \( h \) units right means replacing \( x \) with \( x - h \). Here, \( h = 3 \), so \( x \to x - 3 \), giving \( (x - 3)^2 \).

Step4: Analyze Vertical Shift

A vertical shift of \( k \) units up means adding \( k \). Here, \( k = 1 \), so \( + 1 \), resulting in \( (x - 3)^2 + 1 \).

Step5: Match with Options

Option B describes shifting 3 units right (\( h = 3 \)) and 1 unit up (\( k = 1 \)) to get \( f(x) = (x - 3)^2 + 1 \), which matches our analysis.

Answer:

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