Find all asymptotes, \( x \)-intercepts, and \( y \)-intercepts (if they do not exist, answer "DNE") for the graph of the rational function and sketch the graph of the function. \( f(x)=\frac{4 x}{x^{2}+8 x+16} \)

We are given the function   f(x) = (4x)⁄(x² + 8x + 16). Step 1. Factor the Denominator Notice that   x² + 8x + 16 = (x + 4)². So the function can be rewritten as   f(x) = 4x⁄(x + 4)². The domain is all real numbers except x = –4 (because (–4 + 4)² = 0). Step 2. Find the Vertical Asymptote A vertical asymptote occurs when the denominator is zero (and the numerator is not simultaneously zero – i.e. no cancellation occurs). Since (x + 4)² = 0 when x = –4, there is a vertical asymptote at   x = –4. Step 3. Find the Horizontal Asymptote Compare the degrees of the numerator and denominator:  • Numerator: degree 1 (4x)  • Denominator: degree 2 ((x + 4)²) Since the degree in the numerator is less than that in the denominator, we have a horizontal asymptote at   y = 0. Step 4. Find the x-intercept(s) The x-intercepts occur where f(x) = 0 (and the function is defined). Set the numerator equal to zero:   4x = 0 ⟹ x = 0. Since x = 0 is in the domain, the only x-intercept is   (0, 0). Step 5. Find the y-intercept The y-intercept is found by evaluating f(0):   f(0) = 4(0)⁄(0 + 4)² = 0⁄16 = 0. So the y-intercept is also   (0, 0). Step 6. Sketching the Graph • The domain is x ∈ (–∞, –4) ∪ (–4, ∞). • There is a vertical asymptote at x = –4. • There is a horizontal asymptote at y = 0. • The graph crosses the origin (0, 0), which is both the x– and y–intercept. • Behavior near the vertical asymptote:  – For x-values approaching –4 (from either side), the denominator (x + 4)² is very small and positive, while the numerator 4x is near –16. Thus, f(x) → –∞ on both sides of x = –4. • End behavior:  – As x → +∞, f(x) ≈ 4x/(x²) = 4/x → 0 (from above, if x is positive).  – As x → –∞, f(x) ≈ 4x/(x²) = 4/x → 0 (from below, since x is negative). The graph therefore comes in from near y = 0 on both ends, drops down sharply near the vertical line x = –4, and passes through the origin. Summary of answers:  • Vertical asymptote: x = –4.  • Horizontal asymptote: y = 0.  • x-intercept: (0, 0).  • y-intercept: (0, 0). This information is all you need to sketch the graph of f(x).