Full Solution to #6 on Little AP Assignment- Graphing
6. Sketch the graphs. For each, determine, with algebraic justification, the domain,
range, intercepts, relative extrema (if any). State also the equations of any asymptotes.
Correction: Students asked me if they had to justify the equations of the asymptotes
algebraically to receive full marks. Since you were just asked to STATE them, full marks
are awarded regardless of your (absence of) reasoning. However, in the solution which
follows, I provide this justification, in the event that it is required in the future.
x 2  4x  32
x
2
Domain: x  4x  32  0
a) f (x) 
Let x 2  4x  32  0 to find x intercepts
(x  8)(x  4)  0  x  8, 4 are x intercepts
D f  {x  | x  4 or x  8} see sign diagram below
Most of you found the minimum value of y to take place at
x  16, y 1.09; however, many forgot to include this in the range.
R f  {y  | - 1.09  y  1}
Discussion about the aymptotes:
For horizontal asymptotes, examine
lim f (x)  lim
x
x
= lim
x
x 2  4x  32
x
x 2  4x  4  4  32
x
(x  2)2
x
x
x2
= lim
x
x
=1
However,
= lim
lim f (x)  lim
x
x
x 2  4x  32
x
(x  2)2
x
x
(x  2)
= lim
x
x
= 1
= lim
 y  1 and y  1 are BOTH horizontal asymptotes
Please find the graph shown below.
There are no vertical asymptotes, as x  0 is in a section of the graph which is undefined.
 D f  {x  | x  4 or x  8}
x 2  4x  32
8
-4
f'(x)
Discussion about the aymptotes:
For horizontal asymptotes, examine
lim f (x)  lim
x
x
= lim
x
x 2  4x  32
x
x 2  4x  4  4  32
x
(x  2)2
= lim
x
x
x2
= lim
x
x
=1
However,
lim f (x)  lim
x
x
x 2  4x  32
x
(x  2)2
x
x
(x  2)
= lim
x
x
= 1
 y  1 and y  1 are BOTH horizontal asymptotes
Please find the graph shown below.
= lim
There are no vertical asymptotes, as x  0 is in a section of the graph which is undefined.
b)
g(x)  x 2  4 x  32  x
Horizontal/Oblique Asympotes
lim x 2  4 x  32  x
x
 lim x 2  4 x  4  4  32  x
x
 lim (x  2)2  36  x
x
 (x  2)  x  2
 y  2 is a horizontal asymptote as x  
lim g(x)  (x  2)  x  2x  2
x
 y  2x  2 is an oblique asymptote as x  
The domain and range also caused a problem.
Dg  D f
but to find the range, we need to find the y values of the function at x= - 4 and x= 8
At x=-4, g(-4) = 4 . At x=8, g(8) = -8
 Rg  {y  | 8  y  2 or y  4}
See graph