Expression 16: "c" left parenthesis, "x" , right parenthesis equals "g" Subscript, 3 , Baseline left parenthesis, "x" , right parenthesis Has graph.cx=g3x
Extend to 3D
16
Sample as product of factors:
17
Expression 18: "f" Subscript, 3 , Baseline left parenthesis, "x" , right parenthesis equals "a" left parenthesis, "x" , right parenthesis "b" left parenthesis, "x" , right parenthesis "c" left parenthesis, "x" , right parenthesis Has graph.f3x=axbxcx
Extend to 3D
18
Real-Valued output for complex inputs to g_3 where imaginary part is non-zero. (Accounts for vertical translations but not any horizontal translations, so only works for strictly imaginary-only inputs and when g_3's vertex is on the y axis).
19
Expression 20: left parenthesis, 0 , negative "t" squared plus "k" Subscript, 3 , Baseline , "t" , right parenthesis Has graph.0,−t2+k3,t
domain t Minimum: negative 500−500
less than or equal to "t" less than or equal to≤t≤
domain t Maximum: 500500
20
Expression 21:
21
Complex Roots of Sample:
22
Expression 23: left parenthesis, negative "k" Subscript, 1 , Baseline , 0 , 0 , right parenthesis equals left parenthesis, 1 , 0 , 0 , right parenthesis Has graph.−k1,0,0
equals=
left parenthesis, 1 , 0 , 0 , right parenthesis1,0,0
23
Expression 24: left parenthesis, negative "k" Subscript, 2 , Baseline , 0 , 0 , right parenthesis equals left parenthesis, negative 2 , 0 , 0 , right parenthesis Has graph.−k2,0,0
equals=
left parenthesis, negative 2 , 0 , 0 , right parenthesis−2,0,0
24
If g_3's roots are imaginary:
25
Expression 26: left parenthesis, StartRoot, negative "k" Subscript, 3 , Baseline , EndRoot , 0 , 0 , right parenthesis equals "u" "n" "d" "e" "f" "i" "n" "e" "d" Has graph.−k3,0,0
undefined
26
Expression 27: left parenthesis, negative StartRoot, negative "k" Subscript, 3 , Baseline , EndRoot , 0 , 0 , right parenthesis equals "u" "n" "d" "e" "f" "i" "n" "e" "d" Has graph.−−k3,0,0
undefined
27
If g_3's roots are real:
28
Expression 29: left parenthesis, 0 , 0 , negative StartRoot, "k" Subscript, 3 , Baseline , EndRoot , right parenthesis equals left parenthesis, 0 , 0 , negative 2 , right parenthesis Has graph.0,0,−k3
equals=
left parenthesis, 0 , 0 , negative 2 , right parenthesis0,0,−2
29
Expression 30: left parenthesis, 0 , 0 , StartRoot, "k" Subscript, 3 , Baseline , EndRoot , right parenthesis equals left parenthesis, 0 , 0 , 2 , right parenthesis Has graph.0,0,k3
equals=
left parenthesis, 0 , 0 , 2 , right parenthesis0,0,2
