1. The problem is to analyze the function $$f(x) = \frac{5x}{4 - x^2}$$ and understand its behavior. 2. This is a rational function where the numerator is $5x$ and the denominator is $4 - x^2$. 3. Important rules for rational functions: - The function is undefined where the denominator is zero. - To find vertical asymptotes, solve $4 - x^2 = 0$. - To find horizontal asymptotes, compare degrees of numerator and denominator. 4. Solve for vertical asymptotes: $$4 - x^2 = 0 \implies x^2 = 4 \implies x = \pm 2$$ So, vertical asymptotes at $x = 2$ and $x = -2$. 5. Degree of numerator is 1 (since $5x$), degree of denominator is 2 (since $x^2$ term). Since denominator degree > numerator degree, horizontal asymptote is $y = 0$. 6. Find x-intercept by setting numerator zero: $$5x = 0 \implies x = 0$$ So, x-intercept at $(0,0)$. 7. Find y-intercept by evaluating $f(0)$: $$f(0) = \frac{5 \cdot 0}{4 - 0} = 0$$ So, y-intercept at $(0,0)$. 8. Summary: - Vertical asymptotes at $x = \pm 2$ - Horizontal asymptote at $y = 0$ - Intercept at origin $(0,0)$ Final answer: The function has vertical asymptotes at $x=2$ and $x=-2$, a horizontal asymptote at $y=0$, and passes through the origin.