Download Functions:

Functions
Polynomial Functions:
y=a0+a1x+a2x2+…+anxn
Linear function: n=1
y= a0+a1x;
a0: intercept. It is the y value when x=0.
a1: slope, slope indicates the rate of change in y as x changes
It is one of the most commonly used functions, including environmental modeling. For
example, if assuming you are 100 miles on your way away from home, you are driving at
60 miles/hr, how far away you are from home 3 hours later.
a0=100
a1=60
y= a0+a1x=100+60*3=280.
Quadratic functions: n=2
y=a0+a1x+a2x2
Solution of a quadratic equation:
ax2+bx+c=0
if (b2-4ac)>0, then the equation has a two different real solutions,
x1 
b  b 2  4ac
;
2a
x2 
b  b 2  4ac
2a
if (b2-4ac)=0, then the equation has a two identical real solutions,
x1  x 2 
b
2a
if (b2-4ac)<0, then the equation has no real solutions.
Examples:
x2-5x-6=0;
-3x2+2x-5=0;
4x2-16x+16=0;
Exponential Functions:
y=ax
y=ex=exp(x), where e=2.71828, the base of natural log.
x
a
x
1
1
   x
a
a
ax+y =ax×ay
a x y 
ax
ay
x
ax
a
   x
b
b
(ax)y=ax×y
(ab)x=axbx
(a+b)2=a2+2ab+b2
(a-b)2=a2-2ab+b2
a2 – b2=(a-b)(a+b)
Exponentiation:
a n  a
a.  ...
 a (note: a
a.
 ...

a  ?)




n
n
a0=1
1
a-1=
a
a1  a 
a
0.5
1
1
 1
1 a
a
1
2
a  a
Logarithmic Functions:
y=loga(x), x=ay
100=102, thus, 2=log10100=log100
a=e, y=ln(x)
log(xy)=log(x) + log(y)
log(x/y)=log(x) – log(y)
log(xc)=c*log(x)
logaa=1
loga1=0
logay =
log b y
, e.g. log1001000000=log1000000/log100=6/2=3
log b a
Trigonometric Functions:
sin(x)
cos(x)
tan(x)
ctan(x)
A*sin(ωx+φ): A is amplitude, p=2π/ω, φ=phase.
δ= 23.5×sin(2π*(284+Jday)/365), which as period of 365 day and phase of 2π*284/365.
sin2(x)+cos2(x)=1
tag(x)=1/ctag(x)
tag(x)=sin(x)/cos(x)
sin( x  y )  sin x cos y  cos x sin y
cos( x  y )  cos x cos y  sin x sin y
sin 2x  2 sin x cos x
Degrees and radiances: a circle is 360 in degrees and 2π in radiances, thus
1 degree=π/180 radiances
1 radiance=180/π ≈57.3 degrees
Numerical Exercises
1. plot y=ax+b with x varies from -10 to 10:
(1) where a=-2, -1, 0, 1, 2 and b=5 all in a single plot.
(2) where a=1, b= -2, -1, 0, 1, 2 all in a single plot.
(3) based on the two plots, describe how “a” and “b” control the line.
2. plot y=exp(ax): with x varies from 0 to 10:
(1) a=0.1
(2) a=-0.1
(3) based on the two plots, describe how the sign of “a” control the line.
3. Solve x from the following quadratic equations:
(1) x2+2x +1=0
(2) 5x2-3x+6=0
(3) 5x2-xx-6=0
(4) ax2+3mx+3tx2+2nx+10+c=0
2
(5) e 2 x x  10
4. Evaluate the following expressions without using a calculator unless asked.
(1) Log(100)
(2) log(200)-log2
(3) log100(1000000)
(4) log5(5)
(5) ln(e)
(6) ln(10)
(7) log(-1)
(8) log(1.0)
(9) log(4)+log(25)
(10)
a5+a3
(11)
a5*a3
(12)
a5/a3
(13)
(1)-1
(14)
(-1)2n , where n is a positive integer.
(15)
(-1)2n+1, where n is a positive integer.
(16)
sin(75)
(17)
sin(120)
(18)
Given sin(x)=0.3, what is cos(x)? Please use a calculator to figure
out what the angle of x is in degrees?
2
(19)
2
 
3
2
(20)
2
 
3