To analyze the function $f(x) = \frac{2x^2 + 7x - 15}{3x^2 - 10x + 3}$, let’s determine the required properties step-by-step.

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1. The $y$-intercept

The $y$-intercept is found by evaluating $f(0)$. Substituting $x = 0$:

$f(0) = \frac{2(0)^2 + 7(0) - 15}{3(0)^2 - 10(0) + 3} = \frac{-15}{3} = -5$

So, the $y$-intercept is at the point (0, -5).

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2. The $x$-intercepts

The $x$-intercepts occur where the numerator equals zero, i.e., $2x^2 + 7x - 15 = 0$. Solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 2$, $b = 7$, $c = -15$:

$x = \frac{-7 \pm \sqrt{7^2 - 4(2)(-15)}}{2(2)} = \frac{-7 \pm \sqrt{49 + 120}}{4} = \frac{-7 \pm \sqrt{169}}{4}$

$x = \frac{-7 \pm 13}{4}$

$x = \frac{-7 + 13}{4} = \frac{6}{4} = \frac{3}{2}, \quad x = \frac{-7 - 13}{4} = \frac{-20}{4} = -5$

So, the $x$-intercepts are at the points $\left(\frac{3}{2}, 0\right)$ and $(-5, 0)$.

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3. Vertical asymptotes

Vertical asymptotes occur where the denominator equals zero, i.e., $3x^2 - 10x + 3 = 0$. Solve this quadratic equation using the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here, $a = 3$, $b = -10$, $c = 3$:

$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(3)(3)}}{2(3)} = \frac{10 \pm \sqrt{100 - 36}}{6} = \frac{10 \pm \sqrt{64}}{6}$

$x = \frac{10 + 8}{6} = \frac{18}{6} = 3, \quad x = \frac{10 - 8}{6} = \frac{2}{6} = \frac{1}{3}$

So, the vertical asymptotes are at $x = 3$ and $x = \frac{1}{3}$.

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Final Results:

1. $y$-intercept: $(0, -5)$
2. $x$-intercepts: $\left(\frac{3}{2}, 0\right)$ and $(-5, 0)$
3. Vertical asymptotes: $x = 3$ and $x = \frac{1}{3}$

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Do you have any questions or need further details?
Here are some related questions for further exploration:

1. What are the horizontal asymptotes of this function?
2. How would you graph this function based on the intercepts and asymptotes?
3. What happens to the function as $x \to \infty$ or $x \to -\infty$?
4. Can you confirm the domain of $f(x)$ based on the asymptotes?
5. How can the behavior near the asymptotes be analyzed?

Tip: Always factorize where possible to simplify finding intercepts and asymptotes.