91Ó°ÊÓ

When graphing rational functions, understanding reflections is key. In our exercise, we started with the parent function, which is \(\frac{1}{x}\).
The negative sign in front of the function indicates a reflection across the x-axis. This means every point on the graph of \( \frac{1}{x} \) will be reflected, or flipped, over the x-axis.
This transformation changes the graph from the first and third quadrants to the second and fourth quadrants. So, the function \( \frac{1}{x} \) transforms into \( -\frac{1}{x} \).
This step is essential to correctly interpret both the shape and position of the rational function.

After reflecting a function across the x-axis, we often need to apply vertical stretches or compressions.
In our case, the reflection step yielded \( -\frac{1}{x} \). To apply a vertical stretch by a factor of 3, you multiply the whole function by 3.
This makes our function \( -\frac{3}{x} \).
Stretching vertically means that the graph will move further away from the x-axis if the factor is greater than 1.
So, every y-value of the function will be 3 times farther from the x-axis compared to \( -\frac{1}{x} \). Visually, this transformation makes the graph steeper.
This helps depict how much quicker the values of the function change as x moves away from zero.

Asymptotes are lines that the graph of a function approaches but never actually touches.
For the rational function \( f(x) = -\frac{3}{x} \), there are two important asymptotes to consider: a vertical asymptote and a horizontal asymptote.
The vertical asymptote occurs where the denominator equals zero. For our function, this is at \( x = 0 \).
This means that as x approaches zero from the left or the right, the function's value tends to infinity or negative infinity. Next, the horizontal asymptote indicates the value that the function approaches as x moves towards positive or negative infinity.
For our function, this is y = 0. This is because, as x becomes very large or very small, the value of \( -\frac{3}{x} \) approaches zero but never actually reaches it.
These asymptotes are crucial for understanding the behavior of the graph and are always a significant aspect to plot and consider.