   Chapter 3.5, Problem 11E

Chapter
Section
Textbook Problem

# 1-40 Use the guidelines of this section to sketch the curve. y = x − x 2 2 − 3 x + x 2

To determine

To sketch:

The curve of given function

Explanation

1) Concept:

i) The domain is the set of x values for which the function is defined.

ii) To find x-intercept, put y=0, and to find y-intercept, put x=0 in the given function.

iii) Symmetry: To find symmetry, replace x by –x and check the behaviour of function. Thus, if f-x=fx, then it is an even function, so its curve has y-axis symmetry. If f-x=-fx, then it is an odd function, so it has symmetry about the origin. And if f-x-fxfx, then it has no symmetry.

iv) An asymptote is a tangent at infinity. To find horizontal, vertical, and slant asymptote, follow the rules.

v) A function is increasing if f'x>0  and decreasing if f'x<0 in that particular interval.

vi) The number f(c) is a local maximum value of f  if fcf(x) when x is near c and is a local minimum value of f if fc f(x) when x is near c.

vii) If f''x>0, the function is concave up and if f''x<0, the function is concave down in that particular interval. And if f''x=0, give the values of inflection points.

2) Given:

y=x-x22-3x+x2

3) Calculation:

Here, first find the domain of the given function and the x & y intercepts. Next, check the symmetry, asymptotes, intervals of increase and decrease, local maximum and minimum values, concavity, and points of inflection. Using these, sketch the curve.

A. Domain

Since y=x-x22-3x+x2  is a rational expression, its domain is (-,1)1, 2(2, ).

Because at x=1,2, the denominator becomes 0, it makes the function undefined.

B. Intercepts

For y intercept, plug x=0  in the given function and solve it for y.

y=0-022-30+02

y=0

y  intercept is (0, 0).

For x intercept, plug y=0 in the original function, and solve it for x.

0=x-x22-3x+x2

x=1,0; but x=1 is not in the domain hence

x  intercept is (0, 0).

C. Symmetry

For symmetry, replace each x by -x.

f-x= -x--x22-3-x+-x2

f(-x)=-x-x22+3x+x2

f-xfx -f(x)

The function is neither odd nor even. Therefore, it has no any symmetry.

D. Asymptote

a) Horizontal asymptotes:

limx-  x-x22-3x+x2=-1, limx x-x22-3x+x2=-1

Horizontal asymptote is y=-1.

b) Vertical asymptotes:

fx=x-x22-3x+x2=x1-xx-1x-2=-xx-2

Since, the function becomes undefined at 2, the vertical asymptote is x=2

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