Exponential Functions
Day 3, Dominica
Mrs. Lukowski’s Algebra II
Welcome
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Today we are starting something new.
Say goodbye to anything rational (rational expressions,
rational equations, and rational functions.)
You should be handed your classwork packet for the next
few days. This will be a quiz next Monday when I return.
Each day will have an assignment for you to complete
during class. If you work during class, you shouldn’t need
to do homework.
You are welcome to copy this presentation’s notes into
your own notes for your reference.
Today we will explore the world of exponential functions.
Consider this...
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You brushed your teeth today.You did so well that only
10 dental germs are left in your mouth.
Unfortunately, you cannot brush again for the next 36
hours.
Each dental germ can copy itself every hour.
How could you calculate the total number of dental
germs in your mouth at then end of the 36 hours?
𝐿𝑒𝑡 𝑓 𝑥 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑑𝑒𝑛𝑡𝑎𝑙 𝑔𝑒𝑟𝑚𝑠
𝐿𝑒𝑡 𝑥 = 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 ℎ𝑜𝑢𝑟𝑠 𝑠𝑖𝑛𝑐𝑒 𝑏𝑟𝑢𝑠ℎ𝑖𝑛𝑔
The Plaque Conundrum
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x=0 hr, we have 10 germs.
f(0)=10
x=1 hr, each germ copied itself to make 20 germs
f(2)=20 or 2•10
At time=2 h, each of the 20 copied itself to make 40
germs.
f(4)=40 or 4•10 or 22•10
At time=3 h, each of the 40 copied itself to make 80
germs.
Notice these
numbers are
f(3)=80 or 8•10 or 23•10
the same
The Plaque Conundrum
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If we keep going we notice the total number of germs is
equal to f(x)=10•2x
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Now we can put 36 hours in the function to see how
many germs we have…
f(x)=10•2(36)=687,194,767,360 germs! Ewww!
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The number of germs rapidly increased to a very large
number.
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This is called exponential growth. The value of our
variable is the exponent of the function. In the dental
case, it represented the number of hours since brushing.
Our standard form for this type of function is as follows:
initial value
base
•Asymptote is y=0
•Domain is all real numbers
•Range is y>0
How to tell if it’s an exponential function?
x
0
1
2
3
4
y
5
7
9
11
13
+2
+2
+2
Increasing by a constant of 2
+2
𝑦 = 2𝑥 + 5
x
0
1
2
3
4
y
3
6
12
24
48
+3
+6
x2
+12
x2
+24
x2
Increasing by a multiple of 2
𝑦 = 3(2)𝑥
Comparing Bases
Graph y=2x
x
y
-3
0.125
-2
0.25
-1
0.5
0
1
1
2
2
4
3
8
Graph y=10x
9
8
7
6
5
4
3
2
1
0
-4 -3 -2 -1-1 0
-2
1
2
3
4
The asymptote is y = 0
x
y
-2
.01
-1
.1
0
1
1
10
2
100
12
11
10
9
8
7
6
5
4
3
2
1
0
-4 -3 -2 -1-1 0
-2
Whoa! This
one is much
more steep.
1
2
3
The asymptote is y = 0
4
Exponential Growth and Decay
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If a>0 and b>1
If a>0 and 0<b<1
Exponential growth (blue)
Exponential decay (yellow)
Identify Growth and Decay…
If 0<b<1, then bxb=a smaller number. Try it.
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Multiply 0.8x0.8
0.8x0.8=0.64
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Multiply 2x2
2x2=4
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So y=a(b)x gets smaller if b
is less than 1.
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So y=a(b)x gets bigger is b
is greater than 1.
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y=3(0.8)x is an exponential
DECAY function
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y=0.5(2)x is an exponential
GROWTH function
Try this…is it growth or decay?
1.
y=10(1/2)x
1.
DECAY
2.
y=(3)x
2.
GROWTH
3.
y=0.1(10)x
3.
GROWTH
4.
y=1250(0.3)x
4.
DECAY
5.
y=2(4/3)x
5.
GROWTH
Go Ahead – Work on Day 1 Worksheet
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Please work by yourself or with a neighbor