Decreasing Exponential Function
Author: Jesus M. Cajigas Santiago

Decay

A population of         people, its decreasing by a factor of        every year.  
If t represents the the elapsed years and P(t) the population after t years, 
what function models the population by year?

Solution:

t

P(t)

 =         =

=                     =

=                             =

=                                    =

=                                           =

=                                                   =

With this information we can conclude that

Build Function

Graph Function

A population of         people is decreasing by a factor of          every       years.  
If t represents the elapsed years and P(t) the population after t years, 
what function models the population by year?

Decreasing Exponential Function

a = true
P0 = 300 (slider from 100 to 300, step 25)
r = 0.5 (slider from 0.4 to 0.8, step 0.1)
c = 4 (slider from 1 to 4, step 1)
text2: "" + P0 + ""
text4: "" + r + ""
text6: "" + c + ""
g(x) = (c * x)
text9: "" + (g(1)) + ""
text10: "" + (g(2)) + ""
text11: "" + (g(3)) + ""
text12: "" + (g(4)) + ""
text13: "" + (g(5)) + ""
f(x) = (P0 * r^(x))
text14: "" + (f(0)) + ""
text15: "" + (f(1)) + ""
text16: "" + (f(2)) + ""
text17: "" + (f(3)) + ""
text18: "" + (f(4)) + ""
text19: "" + (f(5)) + ""
text22: "" + P0 + ""
text23: "" + P0 + ""
text24: "" + P0 + ""
text25: "" + P0 + ""
text26: "" + P0 + ""
text27: "" + P0 + ""
h(x) = r^(x)
text29: "(" + (LaTeX[r]) + ")"
text30: "(" + r + ")(" + r + ")"
text31: "(" + r + ")(" + r + ")(" + r + ")"
text32: "(" + r + ")(" + r + ")(" + r + ")(" + r + ")"
text33: "(" + r + ")(" + r + ")(" + r + ")(" + r + ")(" + r + ")"
text45: "" + P0 + ""
text46: "" + P0 + ""
text47: "" + P0 + ""
text48: "" + P0 + ""
text49: "" + P0 + ""
text50: "" + P0 + ""
text51: "(" + r + ")"
text52: "(" + r + ")"
text53: "(" + r + ")"
text54: "(" + r + ")"
text55: "(" + r + ")"
text56: "(" + r + ")"
text57: "" + (g(0)) + "/" + c + ""
text58: "" + (g(1)) + "/" + c + ""
text59: "" + (g(2)) + "/" + c + ""
text60: "" + (g(3)) + "/" + c + ""
text61: "" + (g(4)) + "/" + c + ""
text62: "" + (g(5)) + "/" + c + ""
text63: "P(t)=" + (LaTeX[P0]) + "(" + (LaTeX[r]) + ")^{\frac{t}{" + (LaTeX[c]) + "}}"
i = 0 (slider from 0 to 6, step 1)
A = (-14.903593464351005, 3.3553901965265083)
B = (-0.8686355483677789, 3.371025015040036)
d = Segment(A, B)
C = (-15, 1.4)
D = (-0.8204322805432817, 1.4428943020601426)
e = Segment(C, D)
E = (-15, -0.7)
F = (2.7466095384695173, -0.67804948221774)
j = Segment(E, F)
G = (-15, -2.8)
H = (4.771146787098403, -2.7989932664956227)
k = Segment(G, H)
I = (-15, -4.8)
J = (7.374123249621257, -4.775327247300013)
l = Segment(I, J)
K = (-15, -6.7)
L = (9.736083373021623, -6.607051424630912)
m = Segment(K, L)
M = (-15, -8.8)
N = (12.194450032070986, -8.727995208908794)
n = Segment(M, N)
O = Point(d)
Q = (-12.5, 5.07013)
R = (-12.5, 3.35539)
S = (-12.5, 1.4)
q = Segment(R, S)
T = (-12.5, 1.4)
U = (-12.5, -0.7)
s = Segment(T, U)
V = (-12.5, -0.7)
W = (-12.5, -2.8)
t = Segment(V, W)
Z = (-12.5, -4.8)
A_1 = (-12.5, -2.8)
a_1 = Segment(Z, A_1)
B_1 = (-12.5, -4.8)
C_1 = (-12.5, -6.7)
b_1 = Segment(B_1, C_1)
D_1 = (-12.5, -6.7)
E_1 = (-12.5, -8.8)
c_1 = Segment(D_1, E_1)
text8: "" + (g(0)) + ""
fGrow(x) = (P0 * r^(x / c))
F_1: (0, fGrow(0))
G_1: (c, fGrow(c))
H_1: ((2 * c), fGrow((2 * c)))
I_1: ((3 * c), fGrow((3 * c)))
J_1: ((4 * c), fGrow((4 * c)))
K_1: ((5 * c), fGrow((5 * c)))
start = -3
end: (6 * c)
u = 0 (slider from 0 to 100, step 0.5)
f_1 = Function(fGrow, start, start + ((end - start) / 100 * u))
P(t) = (P0 * r^(t / c))
text3: "" + r + ""
b = Segment(O, Q)