Question

use the figure below to answer the question that follows.
the graph of the quadratic function ( y = g(x) ) with vertex ( (1, 2) ) is shown in the ( xy )-plane above. if ( f(x) = x^2 ), which of the following equations is true?
(a) ( g(x) = 2f(x) + 1 )
(b) ( g(x) = 2f(x - 1) )
(c) ( g(x) = f(x + 2) - 1 )
(d) ( g(x) = f(x - 1) + 2 )

Explanation:

Step1: Recall vertex form of quadratic

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

Step2: Analyze transformations

To get from \(f(x)=x^2\) (vertex \((0,0)\)) to \(g(x)\) (vertex \((1,2)\)):

• Horizontal shift: To move the vertex from \(x = 0\) to \(x = 1\), we replace \(x\) with \(x - 1\) (since shifting right by 1 unit: \(f(x - 1)=(x - 1)^2\)).
• Vertical shift: To move the vertex from \(y = 0\) to \(y = 2\), we add 2 to the function: \(g(x)=f(x - 1)+2\).

Step3: Verify with options

• Option A: \(2f(x)+1 = 2x^2+1\), vertex \((0,1)\), incorrect.
• Option B: \(2f(x - 1)=2(x - 1)^2\), vertex \((1,0)\), incorrect.
• Option C: \(f(x + 2)-1=(x + 2)^2-1\), vertex \((-2,-1)\), incorrect.
• Option D: \(f(x - 1)+2=(x - 1)^2+2\), vertex \((1,2)\), correct.

Answer:

D. \(g(x) = f(x - 1) + 2\)