4. sketch the graph of the following function. clearly identify all intercepts and asymptotes.\n\n$$f(x) =…

Explanation:

Step1: Factor the numerator and denominator

$$f(x) = \frac{4(x^2 - 9)}{(x - 4)(x + 2)} = \frac{4(x - 3)(x + 3)}{(x - 4)(x + 2)}$$

Step2: Find the y-intercept

$$f(0) = \frac{4(0)^2 - 36}{(0)^2 - 2(0) - 8} = \frac{-36}{-8} = 4.5$$

Step3: Find the x-intercepts

$$4(x - 3)(x + 3) = 0 \implies x = 3, x = -3$$

Step4: Find vertical asymptotes

$$(x - 4)(x + 2) = 0 \implies x = 4, x = -2$$

Step5: Find the horizontal asymptote

$$\lim_{x \to \infty} \frac{4x^2 - 36}{x^2 - 2x - 8} = \frac{4}{1} = 4$$

Step6: Sketch the graph

Plot intercepts $(3, 0), (-3, 0), (0, 4.5)$ and asymptotes $x=4, x=-2, y=4$.

Answer:

y-intercept: $(0, 4.5)$ x-intercepts: $(3, 0)$ and $(-3, 0)$ Vertical Asymptotes: $x = 4$ and $x = -2$ Horizontal Asymptote: $y = 4$ The graph consists of three branches: one to the left of $x=-2$ approaching $y=4$ and decreasing toward $-\infty$, one between $x=-2$ and $x=4$ passing through $(-3,0), (0,4.5), (3,0)$, and one to the right of $x=4$ approaching $y=4$ from above.