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To solve this problem, let’s analyze the rational function given:

\[
f(x) = \frac{2x^2 - 9x - 5}{2x^2 + 13x + 15}
\]

1) **Finding the y-intercept:**
   - To find the y-intercept, set \( x = 0 \) in \( f(x) \):
   \[
   f(0) = \frac{2(0)^2 - 9(0) - 5}{2(0)^2 + 13(0) + 15} = \frac{-5}{15} = -\frac{1}{3}
   \]
   - **Y-intercept is at \( \left(0, -\frac{1}{3}\right) \).**

2) **Finding the x-intercepts:**
   - X-intercepts occur when the numerator is zero, so we solve \( 2x^2 - 9x - 5 = 0 \).
   - Factor or use the quadratic formula:
   \[
   x = \frac{-(-9) \pm \sqrt{(-9)^2 - 4 \cdot 2 \cdot (-5)}}{2 \cdot 2} = \frac{9 \pm \sqrt{81 + 40}}{4} = \frac{9 \pm \sqrt{121}}{4} = \frac{9 \pm 11}{4}
   \]
   - This gives \( x = \frac{20}{4} = 5 \) and \( x = \frac{-2}{4} = -\frac{1}{2} \).
   - **X-intercepts are at \( (5, 0) \) and \( \left(-\frac{1}{2}, 0\right) \).**

3) **Finding the vertical asymptotes:**
   - Vertical asymptotes occur where the denominator is zero, so solve \( 2x^2 + 13x + 15 = 0 \).
   - Factor the quadratic:
   \[
   2x^2 + 13x + 15 = (2x + 3)(x + 5) = 0
   \]
   - This gives \( x = -\frac{3}{2} \) and \( x = -5 \).
   - **Vertical asymptotes are at \( x = -\frac{3}{2} \) and \( x = -5 \).**

### Summary of Results
1. Y-intercept at \( \left(0, -\frac{1}{3}\right) \)
2. X-intercepts at \( (5, 0) \) and \( \left(-\frac{1}{2}, 0\right) \)
3. Vertical asymptotes at \( x = -\frac{3}{2} \) and \( x = -5 \)

Would you like more details or have any questions?

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Here are some related questions to explore further:
1. How do we determine horizontal asymptotes for rational functions?
2. Can this function have oblique asymptotes?
3. How do changes in the numerator affect the x-intercepts?
4. What happens to the graph near the vertical asymptotes?
5. How can we determine if there are any holes in this function?

**Tip:** For rational functions, vertical asymptotes occur where the denominator equals zero and the numerator is non-zero.

Let \( f(x) = \frac{2x^2 - 9x - 5}{2x^2 + 13x + 15} \). Find the y-intercept, x-intercepts, and vertical asymptotes of the function.

Learn how to find the y-intercept, x-intercepts, and vertical asymptotes of the rational function \( f(x) = \frac{2x^2 - 9x - 5}{2x^2 + 13x + 15} \) with step-by-step solutions. This problem covers key algebra concepts such as intercepts and asymptotes.

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