1. **Problem statement:** We are given the function $$f(x) = \frac{x^3 - 16x}{-4x^2 + 4x + 24}$$ and asked to analyze and graph it, including its oblique (slant) asymptotes. 2. **Simplify the function:** - Numerator: $$x^3 - 16x = x(x^2 - 16) = x(x-4)(x+4)$$ - Denominator: $$-4x^2 + 4x + 24 = -4(x^2 - x - 6) = -4(x-3)(x+2)$$ So, $$f(x) = \frac{x(x-4)(x+4)}{-4(x-3)(x+2)}$$. 3. **Find vertical asymptotes:** Vertical asymptotes occur where the denominator is zero and numerator is not zero. - Denominator zeros: $$x=3$$ and $$x=-2$$ - Numerator zeros at these points: none (since $$x=3$$ or $$x=-2$$ do not make numerator zero) Thus, vertical asymptotes at $$x=3$$ and $$x=-2$$. 4. **Find horizontal or oblique asymptotes:** Degree numerator = 3, degree denominator = 2, so there is an oblique asymptote. Perform polynomial division of numerator by denominator: Divide $$x^3 - 16x$$ by $$-4x^2 + 4x + 24$$: - Leading term division: $$\frac{x^3}{-4x^2} = -\frac{1}{4}x$$ - Multiply denominator by $$-\frac{1}{4}x$$: $$-\frac{1}{4}x \times (-4x^2 + 4x + 24) = x^3 - x^2 - 6x$$ - Subtract: $$(x^3 - 16x) - (x^3 - x^2 - 6x) = 0 + x^2 - 10x$$ Now divide $$x^2 - 10x$$ by $$-4x^2 + 4x + 24$$: - Leading term division: $$\frac{x^2}{-4x^2} = -\frac{1}{4}$$ - Multiply denominator by $$-\frac{1}{4}$$: $$-\frac{1}{4} \times (-4x^2 + 4x + 24) = x^2 - x - 6$$ - Subtract: $$(x^2 - 10x) - (x^2 - x - 6) = 0 - 9x + 6 = -9x + 6$$ Remainder is $$-9x + 6$$. So the division gives: $$f(x) = -\frac{1}{4}x - \frac{1}{4} + \frac{-9x + 6}{-4x^2 + 4x + 24}$$ 5. **Oblique asymptote:** The oblique asymptote is the quotient without the remainder term: $$y = -\frac{1}{4}x - \frac{1}{4}$$ 6. **Summary:** - Vertical asymptotes at $$x=3$$ and $$x=-2$$ - Oblique asymptote: $$y = -\frac{1}{4}x - \frac{1}{4}$$ 7. **Graph features:** - The function has zeros at $$x=0$$, $$x=4$$, and $$x=-4$$ (from numerator factorization). - The function is undefined at vertical asymptotes. - The function approaches the oblique asymptote for large $$|x|$$. This completes the analysis and description of the function and its asymptotes.