Question

given the graph, obtained from the graph ( f(x)=\frac{1}{x} ) using transformations?
a reflected across the y - axis and shifted up 2 units.
b reflected across the x - axis and shifted left 2 units.
c reflected across the y - axis and shifted down 2 units.
d reflected across the x - axis and shifted right 2 units.
e reflected across the x - axis and shifted right ( \frac{1}{2} ) units.

Explanation:

Step1: Recall Transformations of \( y = \frac{1}{x} \)

The parent function is \( f(x)=\frac{1}{x} \). Its graph is a hyperbola with two branches in the first and third quadrants. The given graph has one branch in the first quadrant (upper) and one in the fourth quadrant (lower), so a reflection across the \( x \)-axis (which changes the sign of \( y \), moving the third - quadrant branch to the fourth) is involved.

Step2: Analyze Horizontal Shift

For a horizontal shift of a function \( y = f(x) \), the transformation is \( y = f(x - h) \), where \( h>0 \) is a shift to the right and \( h < 0 \) is a shift to the left. The vertical asymptote of \( y=\frac{1}{x} \) is \( x = 0 \). From the graph, the vertical asymptote seems to be at \( x = 2 \) (since the right - hand branch is near \( x = 2 \)). For the function \( y=-\frac{1}{x - 2} \) (reflected across \( x \)-axis: \( y=-\frac{1}{x} \), then shifted right 2 units: \( y =-\frac{1}{x - 2} \)), the vertical asymptote is \( x = 2 \). So the shift is 2 units to the right.

Step3: Match with Options

• Option A: Reflection across \( y \)-axis would keep the function in first and fourth (for \( y=\frac{1}{-x}=-\frac{1}{x} \)) but the shift is wrong.
• Option B: Reflection across \( x \)-axis but shift left is wrong.
• Option C: Reflection across \( y \)-axis and shift down is wrong.
• Option D: Reflection across \( x \)-axis (changes \( y=\frac{1}{x} \) to \( y =-\frac{1}{x} \)) and shifted right 2 units (changes \( y=-\frac{1}{x} \) to \( y=-\frac{1}{x - 2} \)) matches the graph.
• Option E: Shift right \( \frac{1}{2} \) units is wrong.

Answer:

D. Reflected across the x - axis and shifted right 2 units.