Question

use transformations of f(x) = \frac{1}{x^2} to graph g(x) = \frac{1}{(x - 3)^2} + 5. select the correct graph. \bigcirc a. graph a \bigcirc b. graph b \bigcirc c. graph c \bigcirc d. graph d

Explanation:

Step1: Analyze the transformation of the function

For the function \(y = f(x)=\frac{1}{x^{2}}\), the transformation to \(y = g(x)=\frac{1}{(x - 3)^{2}}+5\) involves two steps.

1. Horizontal translation:

The general form of a horizontal translation of a function \(y = f(x)\) is \(y=f(x - h)\). For \(y=\frac{1}{(x - 3)^{2}}\), compared to \(y=\frac{1}{x^{2}}\), when \(h = 3\), the graph of \(y=\frac{1}{x^{2}}\) is shifted 3 units to the right.

1. Vertical translation:

The general form of a vertical translation of a function \(y = f(x)\) is \(y=f(x)+k\). For \(y=\frac{1}{(x - 3)^{2}}+5\), compared to \(y=\frac{1}{(x - 3)^{2}}\), when \(k = 5\), the graph is shifted 5 units up.

Step2: Check the key - point

The vertex of the function \(y=\frac{1}{x^{2}}\) is \((0,0)\). After the transformation \(y=\frac{1}{(x - 3)^{2}}+5\), using the rules of translation \((x,y)\to(x + h,y + k)\) (where \(h = 3,k = 5\) for our case), the vertex of \(y=\frac{1}{(x - 3)^{2}}+5\) is \((3,5)\).

Answer:

Assuming that in the options (not fully visible in the text - based description but following the transformation rules), the graph with the vertex at \((3,5)\) (obtained by shifting \(y = \frac{1}{x^{2}}\) 3 units right and 5 units up) is the correct one. If we assume option D (since no other clear visual cues but based on the transformation rules) has the vertex at \((3,5)\), then the answer is D.