Name
February 9, 2009
Algebra 1B review packet
“Exponential review: Day 1” page 1
Exponential function review and practice — Day 1
Test this Friday 2/13. Bring your calculator every day, but especially Friday!
Today’s problems emphasize deciding whether a problem situation is linear or exponential, and
setting up the function that models the problem situation (writing y = ··· and NOW-NEXT
descriptions).
Review notes: setting up linear and exponential models
Linear or exponential? Some ways to decide




If the problem has growth or decay at a steady rate, use a linear function.
If the problem has growth or decay with a steady multiplier, use an exponential function.
If the problem has growth or decay by a percentage rate, use an exponential function.
Make a graph: does it have a linear shape (straight line) or exponential shape (curved)?
Setting up a linear model
y = mx + b;
NEXT = NOW + m starting from b.
The m value represents the rate of growth or decay. (m is positive for growth, negative for decay.)
Graphically, m is the slope.
The b value represents the initial amount or starting value.
Graphically, the y-intercept is at (0, b).
Setting up an exponential model
y = a · bx;
NEXT = NOW · b starting from a.
The a value represents the initial amount or starting value.
Graphically, the y-intercept is at (0, a).
The b value is the multiplier. (b > 1 for growth, 0 < b < 1 for decay.)
Here are ways to find the multiplier from different kinds of problem statements.
For exponential functions, here are some different ways to find the multiplier b:
Given information
a number that’s used for repeated multiplication
what fraction or decimal is kept
the percent that’s kept
a growth rate as a fraction
a growth rate as a percent
a decay rate as a fraction
a decay rate as a percent
an input-output table or a graph
How to find b
b = that number.
b = that fraction or decimal.
b = (% kept, changed to decimal).
b = 1 + that fraction.
b = 1 + (growth % changed to decimal).
b = 1 – that fraction.
b = 1 – (decay % changed to decimal).
Use ExpReg on your calculator, or
b = (any y value)/(the previous y value)
Name
February 9, 2009
Algebra 1B review packet
“Exponential review: Day 1” page 2
Review problems
11. Determine whether each of these growth or decay situations is linear or exponential.
If it’s linear, identify the rate (the slope), and write: linear, m = ···
If it’s exponential, identify the multiplier, and write: exponential, b = ···
a. increasing by 37% each year
b. increasing at a steady rate of 37 per year
c. decreasing by 37% each year
d. decreasing at a steady rate of 37 per year
e. each year, having 37 times as much as in the previous year
f. each year, keeping 37% of the previous year’s amount
12. For each table below, there is either a linear function or an exponential function that fits the
table exactly. Find the y = ··· equation and NOW-NEXT descriptions. (Do not use
regression on your calculator for this problem.)
a.
x
y
0
500
1
460
2
420
y = __________________.
3
380
4
340
5
300
6
260
7
220
NEXT = ___________________, starting from _______.
b.
x
y
0
640
1
320
2
160
y = __________________.
3
80
4
40
5
20
6
10
7
5
NEXT = ___________________, starting from _______.
c.
x
y
0
40
1
47.5
2
55
y = __________________.
3
62.5
4
70
5
77.5
6
85
7
92.5
NEXT = ___________________, starting from _______.
d.
x
y
0
2
1
3
2
4.5
y = __________________.
3
6.75
4
5
6
7
10.125 15.1875 22.78125 34.171875
NEXT = ___________________, starting from _______.
Name
February 9, 2009
Algebra 1B review packet
“Exponential review: Day 1” page 3
13. Write two words describing each of these graphs. One of the words should be either
linear or exponential. The other should be either growth or decay.
14. For each of these problem situations, write a y = ··· equation and a NOW-NEXT description.
a. When Emma was in Kindergarten (think of Kindergarten as “Grade 0”), her parents gave
her a weekly allowance of $1.00. Each time she moved up a grade, this allowance was
increased by $0.50. Write equations for finding Emma’s allowance in Grade x.
y = __________________.
NEXT = ___________________, starting from _______.
b. In year 2000, the ticket price at a movie theater was $7.50. Each year since, the price has
increased by 5%. Write equations for finding the ticket price, where x stands for the
number of years since year 2000.
y = __________________.
NEXT = ___________________, starting from _______.
c. An investor bought $10,000 of stock in a company that turned out to not do very well.
Each year, the investment’s value decreased by 7%. Write equations for finding the value
of the investment after x years.
y = __________________.
NEXT = ___________________, starting from _______.
d. Joe has a big pile of laundry to wash. There are 140 pieces of clothing to be washed.
He can wash 20 pieces in each laundry load. Write equations for finding how many
clothes are left after doing x loads of laundry.
y = __________________.
NEXT = ___________________, starting from _______.
e. 200 students signed up to be members of a club, but only 90% of them actually came to
the first meeting, and the attendance at each subsequent meeting was 90% of the
attendance at the meeting before. Write equations for finding the attendance at meeting
number x of the club.
y = __________________.
NEXT = ___________________, starting from _______.
Name
February 9, 2009
Algebra 1B review packet
“Exponential review: Day 1” page 4
15. Answer these questions about the function f(x) = 1.35 · (0.25)x.
a. Is this a linear function or an exponential function? Tell how you know.
b. Is this a growth function or a decay function? Tell how you know.
c. Suppose the formula f(x) = 1.35 · (0.25)x came from a word problem involving a percent
increase or decrease. Which is it (increase or decrease), and what would the percentage be?
d. Suppose the formula f(x) = 1.35 · (0.25)x came from a word problem involving keeping
a percentage of something. What would be the percentage kept?
e. Which of these is the shape of the graph of f(x)?
f. What are the coordinates of the y-intercept of the graph of f(x)?
Name
February 9, 2009
Algebra 1B review packet
“Exponential review: Day 1” page 5
16. Answer these questions about this NOW-NEXT description:
NEXT = NOW – 5, starting from 300.
a. Which does it describe: a linear pattern or an exponential pattern? Tell how you know.
b. Which does it describe: growth or decay? Tell how you know.
c. Make up a word problem of your own that would have this NOW-NEXT description.
(Try to be creative, not just borrow one of my problem situations.)
17. Answer these questions about this NOW-NEXT description:
NEXT = NOW ·
2
3
, starting from 300.
a. Which does it describe: a linear pattern or an exponential pattern? Tell how you know.
b. Which does it describe: growth or decay? Tell how you know.
c. Make up a word problem of your own that would have this NOW-NEXT description.
(Again, try to be creative.)
1. Many people with diabetes must take frequent injections of a medicine containing insulin,
which helps them process glucose (sugar). Once in the bloodstream, the insulin breaks down
quickly. Here is a graph showing a typical pattern of insulin decrease.
Name
February 9, 2009
Algebra 1B review packet
“Exponential review: Day 1” page 6
a. Which type of function models the decay of insulin in the bloodstream: linear or
exponential? Briefly explain why.
b. Using regression on your calculator, find the y = equation for the best-fit function. (You
can use every third point: (0, 10), (9, 6.5), (18, 4.1), … . You will have to estimate some y
values.)
y=
.
Name
February 9, 2009
Algebra 1B review packet
“Exponential review: Day 1” page 7
c. Suppose that this patient’s doctor needs to know how much insulin is present
after 2.5 minutes, 5 minutes, and 10 minutes. These values aren’t shown on the graph,
but use your equation from part b to estimate them.
minutes since entering bloodstream
2.5
units of insulin
5
10
d. Using numbers found in part b, write a NOW-NEXT description of the insulin decay.
(NEXT means the amount of insulin that would be present 1 minute later, compared to NOW.)
NEXT =
, starting from _____.
e. Medical scientists are often interested in the amount of time it takes for a drug to be reduced
to one-half of its original dose. They call this time the half-life of the drug. Based on the
graph, how many minutes is the half-life of insulin?
f. Suppose a patient received an initial dose of 30 units of insulin instead of 10 units.
Tell how each of the following would change (or if it’s unchanged, say so).
i)
the y = equation
ii)
the NOW-NEXT description
iii)
the output values in an input-output table
iv)
the y-intercept of the graph
v)
the general shape of the whole graph
vi)
the half-life
Name
February 9, 2009
Algebra 1B review packet
“Exponential review: Day 1” page 8
2. Various y = ··· and NEXT = ··· rules are given below. Identify each as one of the following:
linear growth, linear decay, exponential growth, exponential decay, or none of the above.
If you’re unsure, the comparison chart on page 2 might be helpful.
a. y = 5 · (0.4)x
h. y =
x
5
b. y = 5 + 0.4x
i. y =
5
x
c. y = 0.4 + 5x
j. NEXT = 0.4 · NOW
d. y = 0.4 · 5x
k. NEXT = NOW + 0.4
e. y = 0.4 + 5x
l. NEXT = NOW – 5
f. y = 5 – 0.4x
m. NEXT = 5 · NOW
g. y = 0.4 – 5x
3. Here are some facts about the Earth. Rewrite each of the numbers in scientific notation.
The first one is done for you as an example.
a. Earth’s population:
6,215,000,000 people
6.215 · 109
or 6.215 E9
b. Earth’s volume:
1,083,000,000,000,000,000 square kilometers
c. Earth’s land area:
58,969,045,000,000 square meters
d. If you fill a bucket with dirt, the portion of the whole Earth that’s in the bucket:
0.0000000000000000000000016
Name
February 11, 2009
Algebra 1B review packet
Exponential review: Day 1 page 9
4. (Optional!) Hypothermia is a life-threatening condition in which body temperature falls well
below the norm of 98.6F. However, because chilling slows normal body functions, doctors
are exploring ways to use hypothermia as a technique for extending time of delicate
operations like brain surgery.1
a. The following table gives experimental data illustrating the relationship between body
temperature and brain activity.
body temperature (F)
brain activity (% of normal)
50
11
59
16
68
24
77
37
86
52
98.6
100
i) Using an appropriate type of regression on your calculator, find a y = ··· rule that
models the table.
ii) Estimate the level of brain activity at a body temperature of 39F (the lowest
temperature that’s been used in surgery experiments on animals).
iii) Estimate the body temperature at which brain activity would be 75% of normal.
Hint: Look at your calculator’s input-output table for the function you found by regression.
b. The following table gives experimental data illustrating the relationship between body
temperature and safe operating time for brain surgery.
body temperature (F)
safe operating time (minutes)
50
45
59
31
68
21
77
14
86
9
98.6
5
i) Using an appropriate type of regression on your calculator, find a y = ··· rule that
models the table.
ii) Estimate the safe operating time for a body temperature of 39F.
iii) Estimate the body temperature at which safe operating time would be 25 minutes.
USA Today, August 1, 2001, “Surgery’s Chilling Future Will Put Fragile Lives on Ice.” per citation in Core-Plus
Mathematics: Contemporary Mathematics in Context, Course 1, page 357.
1