Transcript 3. - Brookwood High School

Warm-Up
Evaluate the expression without using a calculator.
1.
5–2
ANSWER
2.
3. –3
1
25
2
83
ANSWER
4
3
42
–24
4. State the domain and range of
the function
y = –(x – 2)2 + 3.
domain: all real
numbers; range: y ≤ 3
Exponential Graphs with M & M’s!!!

Cycle
0
# of
m & m’s
1


1
2
3


4

5
Make a t chart as shown
Start with 1 m & m
For each cycle, double the number
of m & m’s you have on your paper
towel (record the number each
time)
Continue until you finish the chart
Plot the points on your graphing
calculator
Look at the graph and use
regression to make the equation.
Now let’s do Exponential Decay!

Cycle
0
# of
m & m’s
32

1

2
3
4
5

Create another t chart this time
starting with 32 m & m’s.
½ the m & m’s each time (You may
eat them when you record your
number.)
Continue the chart.
Plot the points and use the
regression key to come up with the
equation.
Exponential Growth
Functions
4.1 (M3)
P. 130
Vocabulary



Exponential function: y = abx (x is the exponent)
If a>0 and b>1, then it is exponential growth.
 B is growth factor
Asymptote: line a graph approaches but never
touches


basic exponential graphs have 1 asymptote
Exponential Growth Model y = a(1+r)t, where t is
time, a is initial amount and r is the % increase
 1 + r is the growth factor
Graph y = b x for b > 1
EXAMPLE 1
x
Graph y = 2 .
SOLUTION
STEP 1
Make a table of values.
STEP 2
Plot the points from the table.
STEP 3
Draw, from left to right, a smooth curve that
begins just above the x-axis, passes through
the plotted points, and moves up to the right.
EXAMPLE 2
x
Graph y = ab for b > 1
Graph the function.
a.
1
y=
2
x
4
SOLUTION
a.
Plot 0, 1 and (1, 2) .Then,
2
from left to right, draw a
curve that begins just
above the x-axis, passes
through the two points,
and moves up to the
right.
EXAMPLE 2
x
Graph y = ab for b > 1
Graph the function.
b.
y=– 5
2
x
SOLUTION
b.
Plot (0, –1) and 1, – 5 .
2
Then,from left to right,
draw a curve that begins
just below the x-axis,
passes through the two
points,and moves down
to the right.
EXAMPLE 3
Graph y = ab
Graph y = 4 2
x–1
x–h
+ k for b > 1
– 3. State the domain and range.
SOLUTION
Begin by sketching the graph
x
of y = 4 2 , which passes
through (0, 4) and (1, 8). Then
translate the graph right 1 unit
and down 3 units to obtain the
graph of y = 4 2 x – 1 – 3.The
graph’s asymptote is the line
y = –3. The domain is all real
numbers, and the range is
y > –3.
GUIDED PRACTICE
for Examples 1, 2 and 3
Graph the function. State the domain and range.
1.
y = 4x
3.
2.
2
y= 3
3
x
f (x) = 3
x+1
+2
EXAMPLE 4
Solve a multi-step problem
Computers In 1996, there
were 2573 computer viruses
and other computer security
incidents. During the next 7
years, the number of incidents
increased by about 92% each
year.
• Write an exponential growth
model giving the number n of
incidents t years after 1996.
About how many incidents
were there in 2003?
EXAMPLE 4
Solve a multi-step problem
• Graph the model.
• Use the graph to estimate the year when there
were about 125,000 computer security incidents.
SOLUTION
STEP 1 The initial amount is a = 2573 and the percent
increase is r = 0.92. So, the exponential
growth model is:
n = a (1 + r)
t
= 2573(1 + 0.92)t
t
= 2573(1.92)
Write exponential growth model.
Substitute 2573 for a and 0.92 for r.
Simplify.
EXAMPLE 4
Solve a multi-step problem
Using this model, you can
estimate the number of
incidents in 2003 (t = 7) to be
n = 2573(1.92)7 247,485.
STEP 2 The graph passes
through the points
(0, 2573) and
(1,4940.16). Plot a few
other points. Then
draw a smooth curve
through the points.
EXAMPLE 4
Solve a multi-step problem
STEP 3
Using the graph, you can
estimate that the number of
incidents was about 125,000
during 2002 (t 6).
GUIDED PRACTICE
4.
for Example 4
What If? In Example 4, estimate the year in which
there were about 250,000 computer security
incidents.
SOLUTION
2003
GUIDED PRACTICE
5.
for Example 4
x
In the exponential growth model y = 527(1.39) ,
identify the initial amount,the growth factor, and
the percent increase.
SOLUTION
Initial amount: 527
Growth factor 1.39
Percent increase 39%
EXAMPLE 5
Find the balance in an account
FINANCE You deposit $4000 in an account that pays
2.92% annual interest. Find the balance after 1 year if
the interest is compounded with the given frequency.
a.
Quarterly
b.
Daily
EXAMPLE 5
Find the balance in an account
SOLUTION
a.
With interest compounded quarterly, the balance
after 1 year is:
r
A=P 1+ n
nt
0.0292
= 4000 1 + 4
Write compound interest formula.
4 1
P = 4000, r = 0.0292, n = 4, t = 1
= 4000(1.0073) 4
Simplify.
= 4118.09
Use a calculator.
ANSWER
The balance at the end of 1 year is $4118.09.
EXAMPLE 5
b.
Find the balance in an account
With interest compounded daily, the balance after
1 year is:
r nt
Write compound interest formula.
A=P 1+ n
0.0292
= 4000 1 + 365
365 1
P = 4000, r = 0.0292, n = 365, t = 1
= 4000(1.00008) 365
Simplify.
= 4118.52
Use a calculator.
ANSWER
The balance at the end of 1 year is $4118.52.
GUIDED PRACTICE
6.
a.
for Example 5
FINANCE You deposit $2000 in an account that pays
4% annual interest. Find the balance after 3 years if
the interest is compounded daily.
With interest compounded daily, the balance after 3
years is:
ANSWER
$2254.98