We will analyze the rational function $r(x) = \frac{3x^2 + 6}{x^2 - 2x - 3}$ to determine its intercepts and asymptotes.

1. Finding the x-intercept:

To find the x-intercept, we set $r(x) = 0$, which happens when the numerator is zero: $3x^2 + 6 = 0$ x^2 = -2 \quad \text{(no real solutions since x^2 cannot be negative).} Result: There are no $x$-intercepts ($x, y = \text{DNE}$).

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2. Finding the y-intercept:

To find the y-intercept, we evaluate $r(x)$ at $x = 0$: $r(0) = \frac{3(0)^2 + 6}{(0)^2 - 2(0) - 3} = \frac{6}{-3} = -2$ Result: The $y$-intercept is $(0, -2)$.

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3. Finding the vertical asymptotes:

Vertical asymptotes occur where the denominator is zero and the numerator is nonzero. Solve for the zeros of the denominator: $x^2 - 2x - 3 = 0$ Factorize: $(x - 3)(x + 1) = 0$ $x = 3, \quad x = -1$ Result: The vertical asymptotes are $x = 3$ and $x = -1$.

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4. Finding the horizontal asymptote:

For the horizontal asymptote, compare the degrees of the numerator and the denominator:

• The degree of the numerator ($3x^2 + 6$) is 2.
• The degree of the denominator ($x^2 - 2x - 3$) is 2.

When the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients: $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}} = \frac{3}{1} = 3$ Result: The horizontal asymptote is $y = 3$.

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Final Answers:

• x-intercept: DNE
• y-intercept: $(0, -2)$
• Vertical asymptote: $x = 3, x = -1$
• Horizontal asymptote: $y = 3$

Would you like further clarification or a graph of the function?

5 Relative Questions:

1. How do vertical asymptotes influence the behavior of a function near them?
2. What happens to the graph as $x \to \infty$ or $x \to -\infty$ for rational functions?
3. How do we determine the presence of slant asymptotes instead of horizontal asymptotes?
4. Why is the numerator important when finding the x-intercept of a rational function?
5. Can a rational function have more than one horizontal asymptote?

Tip:

Always verify the domain of the function to ensure no points are excluded when solving for intercepts and asymptotes.