MATH 221

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 13 August 2014

Lecture 16

Graph f(x)=x2-1x3-4x.

Notes:

(a) y= x2-1x3-4x= (x+1)(x-1) x(x2-4) = (x+1)(x-1) x(x+2)(x-2) .
(b) If x=1 then y=0.
(c) If x=-1 then y=0.
(d) If x=2+ then y. (pos·pospos·pos·pos)
(e) If x=2- then y-. (pos·pospos·pos·neg)
(f) If x=0+ then y. (pos·negpos·pos·neg)
(g) If x=0- then y-. (pos·negneg·pos·neg)
(h) If x=-2+ then y. (neg·negneg·pos·neg)
(i) If x=-2- then y-. (neg·negneg·neg·neg)
(j) y=x2-1x3-4x is the same as (-y)=(-x)2-1(-x)3-4(-x) so if we flip y to -y and x to -x the graph stays the same.
(k) As x then y0+.
(l) As x- then y0-.
x y -1 -2 1 2

Graph x+y=1.

Notes:

(a) If we switch x and y this graph stays the same.
(b) If x=0 then y=1, so y=12=1.
(c) If y=0 then x=1.
(d) If x=y then x+x=1 and x=12, x=14.
(e) This graph should be similar to x2+y2=1 or x=y+1.
x y 1 1 22 22 x2+y2=1 x y 1 1 12 12 x+y=1 x y 1 1 14 14 x+y=1

Graph x2-1x2+1=f(x).

Notes:

(a) y= x2-1x2+1= x2+1-2 x2+1 =1-2x2+1.
x y 1 y=1x2+1 Notes:
(a) If x=0, y=102+1=11=1.
(b) If x then y0+.
(c) If x- then y0+.
(d) This graph stays the same if we flip x to -x, y=1x2+1=1(-x)2+1.
x y -2 y=-2x2+1 x y -1 1 y=1-2x2+1 = x2-1 x2+1

Graph ln(4-x2)=f(x).

Notes:

(a) y=ln(4-x2)=ln((2+x)(2-x))=ln(2+x)+ln(2-x).
(b) If we flip x to -x the graph stays the same since y=ln(4-x2)=ln(4-(-x)2).
x y 1 y=lnx x y -1 y=ln(-x) x y -2 1 -1 ln 3 ln 2 y=ln(x+2) x y -1 1 2 ln 3 ln 2 y=ln(2-x) x y -2 2 1 -1 ln 3 2ln 2 ln 2 y=ln(4,x2)=ln(2+x)+ln(2-x).

Graph y=x23(6-x)13.

Notes:

(a) If x=0 then y=0.
(b) If x=6 then y=0.
(c) If x then yx23(-x)13=-x.
(d) If x- then y (y-x again).
x y -1 1 -1 1 y=x3 x y -1 1 -1 1 y=x13
y3=x
x y 1 -1 1 y=x23
=(x13)2
x y 1 -1 -1 1 y=x13
x y 1 -1 5 6 7 y=(6-x)13 6 x y y= x23 (6-x)13 y=-x

Notes and References

These are a typed copy of Lecture 16 from a series of handwritten lecture notes for the class MATH 221 given on October 13, 2000.

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