Question

question: 12
the graph below shows the function ( g(x) = x^2 ) and a transformed function. which of the following accurately describes the transformation?
graph of parabolas
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 + 3)^2 + 1 )
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 + 1)^2 + 3 )

Explanation:

Step1: Recall Vertex Form Transformations

The vertex form of a parabola is \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( g(x) = x^2 \), the vertex is \((0, 0)\). A horizontal shift right by \( h \) units changes \( x \) to \( x - h \), and a vertical shift up by \( k \) units adds \( k \) to the function.

Step2: Analyze the Shifted Vertex

From the graph (and the options), the shifted vertex should be 3 units right (so \( h = 3 \)) and 1 unit up (so \( k = 1 \)). Thus, the function should be \( f(x) = (x - 3)^2 + 1 \), which matches option B. Option C has \( (x + 3)^2 \), which is a shift left, so it's incorrect. Options A (k=5) and D (incorrect shift) are also wrong.

Answer:

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