*Show all work (or provide a brief explanation) clearly, and in order, to receive full credit for your answer. Partial or no credit may be awarded if we cannot see how you arrived at your answer, even if it is correct. Keep your answers brief. A function f(x) and its first two derivatives are given. Use them to find the requested characteristics of f(x). If you find that f does not have something asked for, then please clearly write NONE (or some other indication of this). Explain your decision- do NOT leave it blank or you will receive no credit. Use Calculus to sketch the graph of f (x). f)Find the equation of the vertical asymptote(s). Determine the behavior on both sides of each vertical asymptote. Show your reasoning/work in the space provided. 8(3x? + 4) (x² – 4)3 -8x f(x) = 4 x² . f'(x) = f"(x) = - %3D (x² – 4)² g) Determine where f is increasing, decreasing, and indicate the coordinates of the local maximum and minimum. a) Domain of f (x) h) Determine where f is concave up, concave down, and indicate the inflection points (coordinates). i) Sketch the graph of the function f(x) b) y-intercept coordinate c) x- intercept coordinate(s) (if any) d) Analyze the symmetry of f by determining whether f is even, odd, or neither. Show your reasoning in the space provided. e)Find the equation of the horizontal asymptote(s). Show your reasoning/work in the space provided.

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Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as: