The graph shown in the image appears to be of a rational function with a vertical asymptote and a horizontal asymptote.

Observations:

• There is a vertical asymptote at $x = 2$, as the graph approaches infinity and negative infinity at $x = 2$.
• There is a horizontal asymptote at $y = 0$, which suggests that the function's degree in the denominator is higher than or equal to the numerator.
• The graph looks like a hyperbola, with one part of the curve in the first quadrant and the other part in the third quadrant.
• The function likely takes a form like $y = \frac{a}{x - 2} + b$, where $a$ and $b$ are constants.

From these observations, the most straightforward equation that fits the given graph would be:

$y = \frac{1}{x - 2}$

This equation has:

• A vertical asymptote at $x = 2$ (because $x - 2 = 0$ causes the function to go to infinity).
• A horizontal asymptote at $y = 0$ as $x \to \infty$, which matches the graph.

Do you want more details on how I derived this equation, or do you have any questions?

Relative Questions:

1. What happens to the graph of the function $y = \frac{1}{x - 2}$ as $x$ approaches 2?
2. How would the graph change if we added a constant to the function, like $y = \frac{1}{x - 2} + 1$?
3. What role does the numerator play in a rational function's shape and behavior?
4. How does the degree of the numerator and denominator affect the asymptotes of rational functions?
5. What would happen to the vertical asymptote if we changed the equation to $y = \frac{1}{x - 3}$?

Tip:

To determine the vertical asymptote in a rational function, set the denominator equal to zero and solve for $x$.