Let's analyze the given function and determine its features.

The function is: $f(x) = \frac{-5x + 15}{2x^2 - 18}$

Step 1: Horizontal Asymptote

To determine the horizontal asymptote, we compare the degrees of the polynomial in the numerator and the denominator:

• The degree of the numerator is 1.
• The degree of the denominator is 2.

Since the degree of the denominator is higher, the horizontal asymptote is: $y = 0$

Step 2: Vertical Asymptote

To find the vertical asymptote(s), set the denominator equal to zero and solve for $x$: $2x^2 - 18 = 0$ $x^2 = 9$ $x = \pm 3$ Thus, there are vertical asymptotes at: $x = 3 \quad \text{and} \quad x = -3$

Step 3: $x$-Intercept

The $x$-intercept occurs when $f(x) = 0$, which means the numerator must be zero: $-5x + 15 = 0$ $x = 3$ So, the $x$-intercept is at: $(3, 0)$

Step 4: $y$-Intercept

To find the $y$-intercept, evaluate $f(0)$: $f(0) = \frac{-5(0) + 15}{2(0)^2 - 18} = \frac{15}{-18} = -\frac{5}{6}$ So, the $y$-intercept is at: $\left(0, -\frac{5}{6}\right)$

Step 5: Holes

Holes occur when there is a common factor in the numerator and denominator. Since $-5x + 15$ and $2x^2 - 18$ do not share any factors, there are no holes in the graph of this function.

Summary of Results

• Horizontal Asymptote: $y = 0$
• Vertical Asymptote: $x = 3$ and $x = -3$
• $x$-Intercept: $(3, 0)$
• $y$-Intercept: $\left(0, -\frac{5}{6}\right)$
• Hole: None

Would you like a deeper explanation on any of these points, or have further questions?

1. How can we verify the asymptotes of rational functions using limits?
2. Why are holes in rational functions associated with common factors?
3. How does the degree of the polynomial in the numerator affect the horizontal asymptote?
4. What happens to the graph of a function near its vertical asymptote?
5. How do you find the range of rational functions like this?

Tip: For rational functions, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always $y = 0$.