Question

Which of the following graphs represents a rational function would look like

Answer 1

To determine how a rational function is represented graphically, consider the general features of rational functions. A rational function is typically expressed in the form \( f(x) = \frac{P(x)}{Q(x)} \), where \( P(x) \) and \( Q(x) \) are polynomials. Here are some characteristics you might look for in the graph of a rational function:

1. **Asymptotes**: Rational functions often feature vertical asymptotes where the denominator \( Q(x) = 0 \), which results in undefined values (the graph approaches the line but never touches it). There may also be horizontal or oblique (slant) asymptotes depending on the degrees of the numerator and denominator.

2. **Intercepts**: The graph can cross the x-axis at points where the numerator is zero (i.e., \( P(x) = 0 \)). The y-axis intercept occurs at \( f(0) = \frac{P(0)}{Q(0)} \) as long as \( Q(0) \neq 0 \).

3. **Behavior at infinity**: Depending on the degrees of \( P(x) \) and \( Q(x) \), the end behavior of the function can be characterized. If the degree of \( P \) is less than that of \( Q \), the horizontal asymptote will be at \( y = 0 \). If they are the same degree, the horizontal asymptote will be at the ratio of the leading coefficients.

4. **Discontinuities**: The graph may contain holes if there are common factors in the numerator and denominator that cancel out.

If you have specific graphs to compare, please describe them or provide more details so I can help identify which one represents a rational function.