Question

transforming the graph of a rational function
below is the graph of $y = \frac{1}{x}$.
transform it to make the graph of $y = -\frac{1}{x} - 2$.

Explanation:

Step1: Reflect over x - axis

The original function is \( y=\frac{1}{x} \). For the transformation to \( y = -\frac{1}{x}-2 \), first, we apply the reflection over the x - axis. The rule for reflecting a function \( y = f(x) \) over the x - axis is \( y=-f(x) \). So, reflecting \( y = \frac{1}{x} \) over the x - axis gives us \( y=-\frac{1}{x} \). This changes the sign of the y - values of the original function. For example, if a point \( (a,\frac{1}{a}) \) is on \( y = \frac{1}{x} \), after reflection over the x - axis, it becomes \( (a,-\frac{1}{a}) \).

Step2: Vertical shift down by 2 units

The next transformation is a vertical shift. The rule for a vertical shift of a function \( y = f(x) \) by \( k \) units down is \( y=f(x)-k \). Here, we have \( y = -\frac{1}{x} \) and we want to shift it down by 2 units. So, we subtract 2 from the function: \( y=-\frac{1}{x}-2 \). This moves every point \( (a,-\frac{1}{a}) \) on \( y = -\frac{1}{x} \) down by 2 units to \( (a,-\frac{1}{a}-2) \).

To perform these transformations on the graph:

1. For the reflection over the x - axis: Take the two branches of the hyperbola \( y=\frac{1}{x} \) (the one in the first quadrant and the one in the third quadrant) and flip them over the x - axis. The branch in the first quadrant (where x>0 and y>0) will move to the fourth quadrant (x>0 and y<0), and the branch in the third quadrant (x<0 and y<0) will move to the second quadrant (x<0 and y>0).
2. For the vertical shift down by 2 units: Move each point on the reflected graph (from step 1) down by 2 units. So, the asymptotes of the function \( y = -\frac{1}{x} \) (which are the x - axis \( y = 0 \) and y - axis \( x = 0 \)) will also be shifted. The horizontal asymptote \( y = 0 \) of \( y=-\frac{1}{x} \) will shift down to \( y=-2 \), and the vertical asymptote \( x = 0 \) remains the same.

Answer:

To transform the graph of \( y=\frac{1}{x} \) to \( y = -\frac{1}{x}-2 \), first reflect the graph of \( y=\frac{1}{x} \) over the x - axis, then shift the resulting graph down by 2 units.