Question

11. As a general overview, these are the details one needs to look for when sketching curves (in no particular order: a Domain and range f. Relative extrema (maximum and minimum b. x-and y-intercepts values) c. Vertical asymptotes 9. Absolute extrema (maximum and minimum d. End behavior (including horizontal or values) polynomial asymptotes) h. Intervals of concavity e Intervals of increasing and decreasing i. Points of inflection behavior j. Other points of interest (corners, cusps, vertical tangents, discontinuities, etc) x3-3x +3x-1 Using the guidelines listed above sketch the following curve: y = x2+x-2

Answer 1

Function is (x^3-3x^2+3x-1)/(x^2+x-2) = (x^3-3x^2+3x-1)/(x-1)(x+2), function is defined for all x except at x=-2 . So, domain is R-{-2}.

Y-intercept , f(0)= 0-1/0-2 = 0.5 , so y-intercept is at (0,0.5). No x intercept.

Now, the first derivative of f(x) is f'(x) = (x^4+2x^3-12x^2+14x-5)/(x+2)^2 (x-1)^2.

Putting f'(x)=0 , we found x=-5, 1

f'(-6) = 0.438 and f'(-4)= -1.25 , it change sign from + to -, so there is a minimum turning point at x= -5.

f(-5) =-12, turning point (-5,-12)

Other turning point is (1,0)

There is no inflection point as , f"(x) has no root since -2 satisfy f"(x) but it is in domain gap.

Asymptote : at x= -2, f(x) = -infinity, so there is a vertical asymptote near -2.

The figure is as

-10 5 -10 O -5 5 10 --5- ---10 ---1-5