aim: exponential functions course: alg. 2 & trig. aim: what is an exponential function? do now:...
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Aim: Exponential Functions Course: Alg. 2 & Trig.
Aim: What is an exponential function?
Do Now:Solve/evaluate: 64 = 23n + 1
3 7 25 3r s t
734 x3
4 5
Rewrite in exponential form
Aim: Exponential Functions Course: Alg. 2 & Trig.
Zero Power Property
Properties of Exponents
Product of Powers Property
a0 = 1
Power of Power Property
Power of Product Property
Negative Power Property
Quotients of Powers Property
Power of Quotient Property
am • an = am+n
(am)n = am•n
a-n = 1/an, a 0
(ab)m = ambm
am
an am n , a 0
(a
b)m
am
bm , b 0
Aim: Exponential Functions Course: Alg. 2 & Trig.
Types of exponents
Positive Integer Exponent an = a • a • a • • • • a
n factors
Zero Exponent a0 = 1
Rational Exponent 1/n
a1 n an
Rational Exponent m/n
am n ( an )m
Negative Exponent - m/n
a m n 1
am n
If a 0, am n ( an )m
results in a nonreal number
ex . ( 2)3 2 imaginary number
beware! -23/2 is not the same as (-2)3/2
Aim: Exponential Functions Course: Alg. 2 & Trig.
Exponential Function - Outbreak
Disease Control - “Outbreak”
y = 2x
Aim: Exponential Functions Course: Alg. 2 & Trig.
Exponential Function
y = a • bx
. . have variables as exponents, andwhere a 0, base b > 0, and also b 1.
The x-axis is a horizontal asymptote: ax 0 as x -
The domain (x) is the set of real numbers: (-, )The range (y) is the set of positive real numbers: (0, )
If b > 1, the graph is increasing and continuous
As b increases in value from 1, the slope ofthe graph gets steeper
y = 1 • 2x
y = 2x
When a = 1, graph always goes through (0,1)
(0,1)
What happens as b increases in value?
Aim: Exponential Functions Course: Alg. 2 & Trig.
4
2
-2
-4
-5 5
The b Affect
y = 2xy = (1/2)x
(0,1)
the graph is decreasing - Decay
If b is a positive number other than 1, the graphs of y = bx and y = (1/b)x
are reflections through the y-axis of each other
If b > 1, the graph is increasing - Growth
What if b = 1? horizontal line: y = 1
y = 1
y = a • bx
If 0 < b < 1
Aim: Exponential Functions Course: Alg. 2 & Trig.
4
2
-2
-4
-5 5
f x = 2x
The b Affect: b < 0
y = a • bx
Let b = (-2)?
What if b < 0?
if b < 0, no longer the exponential function
y = 1 • (-2)x
graph: table: x = 1table: x = .1
Aim: Exponential Functions Course: Alg. 2 & Trig.
y = a • bx
The a Affect
(0,1) a = 1
(0,2) a = 2
(0,4)a = 4
y = 2x
y = (2)2x
y = (4)2x
Graph: y = 2x; y = (2)2x; y = (4)2x
Aim: Exponential Functions Course: Alg. 2 & Trig.
4
2
-2
-4
-5 5
f x = 2x
4
2
-2
-4
-5 5
f x = -2x
The a Affect
Graph f(x) = -(1/2)x = -1 • (1/2)x
Graph f(x) = -6x = -1 • (6)x
4
2
-2
-4
-5 5
g x = -1
2
x
4
2
-2
-4
-5 5g x = -6x (0,-1)
y = a • bx
Graph f(x) = -2x = -1 • 2x
Aim: Exponential Functions Course: Alg. 2 & Trig.
The a affect
Graph the exponential functions
11,
5 552
1,6
xx x
yy y
10
5
-5
-10
-10 10
h x = 1
5
x10
5
-5
-10
-10 10
h x = 61
5
x
10
5
-5
-10
-10 10
h x = -21
5
x
Aim: Exponential Functions Course: Alg. 2 & Trig.
Model Problem
Write an exponential function y = abx for a graph that includes (2, 2) and (3, 4)
xy ab 22 ab 2
2a
b
2
2 xy bb
32
24 b
b
3 3 22
24 2 2b b b
b
b = 2
2
2
2a a = 1/2 1
22
xy
general form substitute (2,2) for x & ysolve for a
substitute for a substitute (3,4) for x & y
simplify & solve for b
substitute & solve for a substitute for a and b
Aim: Exponential Functions Course: Alg. 2 & Trig.
6
4
2
-2
-5 5
r x = 1+1
x x
Where’d e Come From?
Graph x
f xx
11
6
4
2
-2
-5 5
y 2.7183
y 2.7183 is asymptotic to f(x).
x = 1 x = 100 x = 10000
Aim: Exponential Functions Course: Alg. 2 & Trig.
4
3
2
1
-1
-2 2
t x = ex
Natural Exponential Function
f(x) = ex
f(x) = (2.71828. . .)x
Evaluate e2
e-1
e0.48
= 7.389056099
= 0.3678794412
= 1.616074402
10
8
6
4
2
-2
-5 5 10
q1 x = 2e0.24x
Graph f(x) = 2e0.24x Graph f(x) = (1/2)e-0.58x
10
8
6
4
2
-2
-5 5 10
r1 x = 1
2e-0.58x
Aim: Exponential Functions Course: Alg. 2 & Trig.
Transforming Functions
If k and h are positive numbers and f(x) is a function, then
• f(x) + k shifts f(x) up k units• f(x) – k shifts f(x) down k units
• f(x + h) shifts f(x) left h units• f(x – h) shifts f(x) right h units
f(x) = (x + h)2 + k - parabolic f(x) = |x + h| + k - absolute value
ex. f(x) = (x – 4)2 + 4 is the image of g(x) = x2 after a shift of 4 units to the right and four units up or a translation of T4,4.
NOTE: k is not the y-intercept.what is the y-intercept for f(x) = (x – 4)2 + 4?
Aim: Exponential Functions Course: Alg. 2 & Trig.
10
5
-5
-10
-10 10
h x = 4x
Transforming the Exponential Function
Graph the exponential functions y = 4x, y = 4x + 2, y = 4x – 3
10
5
-5
-10
-10 10
h x = 4x10
5
-5
-10
-10 10
q x = 4x-3
h x = 4x+2
Aim: Exponential Functions Course: Alg. 2 & Trig.
6
4
2
-2
-4
-6
-5 5
u x = 2x
(0, 1)
6
4
2
-2
-4
-6
-5 5
u x = 2x+3
(-3, 1)
Transforming the Exponential Function
Graph the exponential functions y = 2x+3, y = 2x-2, y = 2x–1 – 2
6
4
2
-2
-4
-6
-5 5
v x = 2x-2
(2, 1)
6
4
2
-2
-4
-6
-5 5
w x = 2x-1-2
(1, 1)
Aim: Exponential Functions Course: Alg. 2 & Trig.
Transforming the Exponential Function
Write the equation and graph the exponential function f(x) = 2x after a
reflection in both the x- & y-axes, and a vertical translation of 4.
Graph y = (1/2)x
Reflect thru x-axis: y = -(1/2)x
Shift up 4 units y = -(1/2)x + 4
4
2
-2
-4
-5 5
f x = 2x
f x = 1
2
x
f x = -1
2
x
f x = -1
2
x+4
Graph y = 2x
Aim: Exponential Functions Course: Alg. 2 & Trig.
Transforming the Exponential Function
Graph y = 2x
Reflect thru x-axis: y = -2x
Shift right 2 units y = -(1/2)x –2
6
4
2
-2
-4
-6
-5 5
h1 x = 2x
6
4
2
-2
-4
-6
-5 5
h1 x = -2x
6
4
2
-2
-4
-6
-5 5
h1 x = -2x-2
Shift up 2 units y = -(1/2)x –2 + 2
6
4
2
-2
-4
-6
-5 5
h1 x = -2x-2+2
Write the equation and graph the exponential function after a reflection in the x-axis, vertically stretched by a factor
of 2, a vertical translation of 2 and a horizontal of 2.
Aim: Exponential Functions Course: Alg. 2 & Trig.
Transforming the Exponential Function
The graph of y = (1/4)x is translated 5 units upward, 8 units to the right and then
reflected in the x-axis. What’s the equation?10
8
6
4
2
-2
-4
-6
-8
5 10 15
f x = 0.25x
g x = 0.25x+5
h x = 0.25x-8+5
r x = - 0.25x-8+5
f(x) = -f(x)
Aim: Exponential Functions Course: Alg. 2 & Trig.
Transforming the Exponential Function
The graph of y = (1/4)x is translated 5 units upward, 8 units to the right and then
reflected in the x-axis. What’s the equation?10
8
6
4
2
-2
-4
-6
-8
5 10 15
f x = 0.25x
g x = 0.25x+5
h x = 0.25x-8+5
r x = - 0.25x-8+5
f(x) = -f(x)