FUNCTIONS
p. 1
8.4 Model situations and relationships using a variety of basic functions (linear, quadratic, logarithmic, exponential, and reciprocal) and piecewise-defined functions.
High School
Community College
EWU
a. Choose a function suitable for modeling a real world situation presented using words or data.
Core 1
Core 2
What rule would relate time and diver’s height above
the water if the tower were 20 meters above the pool
and travel a distance of 4.9t 2 meters in t seconds?
The intensity of sound can be measured by several
different scales. One that uses units of power is watts
per square meter. It is a measure of the pressure that a
sound forces on your ear.
The intensity of sound from a stereo system is a
function of the listener’s distance from the speakers.
Displayed in the table below are some measurements
taken at various distances from speakers of a particular
stereo system.
Sound Intensity table omitted due to space restrictions





Describe the overall pattern relating distance D
and intensity I in these data.
Make a scatterplot of the (D,I) data pairs, and
explain how the shape of the graph matches the
pattern in the data table.
Which of he following movements will cause the
greater decrease in sound intensity? Moving from
1 meter to 2 meters away from the speakers. Or
Moving from 4 meters to 5 meters away from the
speakers.
Experiment with various rules in the functions list
of your graphing calculator or computer software
to find a good model of the relations between I and
Express intensity as a function of distance. If
necessary, look back at Activity 5 about light
SFCC – Beginning & Intermediate Algebra
A cell phone costs 50$ a month plus a one time
activation fee of 35$. How much will this plan cost for
2 years?
SCC – Intermediate Algebra
It is in our curriculum, but I skip it
Basic Algebra
An average person burns about 2500 calories per day
doing normal activities, and 3.7 calories per minute
walking at a moderate pace. Write a function rule that
would determine the total number of calories burned
per day if a person does all of her normal activities, and
walks w minutes during the day.
Intermediate Algebra
Brett, a tile setter, creates a granite design for a
customers foyer. His overhead on the job is $49.00
and the materials cost $5.25 per square foot. He
charges his customer $41.25 per square foot. Write a
polynomial P(x) that represents his profit if the shape
of the design is a square with side x feet
Algebra Concepts
A rectangular playground is to be fenced off and
divided in two by another fence parallel to one side of
the playground. Four hundred feet of fencing is used.
Find the dimensions of the playground that maximize
the total enclosed area. What is the maximum area?
Precalculus I
Construct a function that models the following
situation: the income tax for a given state is computed
at a rate of 5% for the first $20,000 earned and 10% for
anything over that amount.
A rectangular box with volume 60ft3 has a square base.
Find a function that models its surface area in terms of
the length of one side of its base.
FUNCTIONS
p. 2
8.4 Model situations and relationships using a variety of basic functions (linear, quadratic, logarithmic, exponential, and reciprocal) and piecewise-defined functions.
High School
intensity for some clues.
Core 3
Integrated 3
Algebra 2, CPM
In 1912, Japan gave the United States several thousand
flowering cherry trees as a symbol of friendship.
Similarly, the nation of Cameroon plans to give
flowering Satta trees to other countries this year.
When asked how to decide which Satta trees make
good gifts, Cameroon’s chief arborist explained, “We
plant Satta trees when they are 6cm tall and they grow
9cm every year. The trees only flower when they are
taller than 150cm.”
It is very important that the trees Cameroon gives
flower this year. It would be considered an insult to
give a tree that did not bloom. Luckily Cameroon has
many groves of Satta trees from which to select its
gifts. How old must the trees be so that they will
flower within a year?
a) Discuss with your study team whether an inequality
or an equation is appropriate for this situation. Be
prepared to share your reasoning.
b) Write and solve a mathematical sentence to
determine how old the trees can be so they flower this
year.
c) Later, the arborist added, “I almost forgot to tell
you! When the trees become very old, they stop
flowering. Make sure you choose trees that are no
more than 240cm tall.”
Discuss with your team how you can use this additional
information to make sure you choose trees that will
flower. Be prepared to share your answer with the
Community College
EWU
FUNCTIONS
p. 3
8.4 Model situations and relationships using a variety of basic functions (linear, quadratic, logarithmic, exponential, and reciprocal) and piecewise-defined functions.
High School
class.
The Alpine Music Club is going on its annual music
trip. The members of the club are yodelers and they
like to play the xylophone. This year they are taking
their xylophones on a gondola to give a performance
on the top of Mount Monch. The gondola conductor
charges $2 for each yodeler and $1 for each xylophone.
It costs $40 for the entire club including the
xylophones to ride the gondola. Two yodelers can
share a xylophone so the number of yodelers on the
gondola is twice the number of xylophones. How
many yodelers and how many xylophones are on the
gondola?
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Currently a junior, I.M. Smart has completed 30
courses at Ferris and now has a cumulative GPA of
3.35. How many additional courses will I.M. have to
get A’s in to raise his GPA to some desired value?
Define a function g c  which determines the GPA g,
when given the number of additional courses taken, c.
Define your variables in your own words.
A square of side length x is cut from each corner of a
10 inch by 14 inch piece of cardboard and the sides
folded up to make an open topped box. Write the
volume of the box as a function of x . What is the
implied domain of this function? What is the relevant
domain for the problem situation? What dimensions of
the box will give the maximum volume?
Community College
EWU
FUNCTIONS
p. 4
8.4 Model situations and relationships using a variety of basic functions (linear, quadratic, logarithmic, exponential, and reciprocal) and piecewise-defined functions.
High School
Community College
EWU
CBS Television is ready to produce the next season of
Survivor, aptly named Survivor Spokane. The first part
of the competition involves all 30 participants
surviving in a math classroom for an entire semester.
Unfortunately, one of the 30 participants arrived with
the highly contagious disease pre-calcula dementia. In
fact, by the third day, there were already a total of 6
people infected. Find a logistical model for the spread
of the disease. Be sure to define your variables.
b. Determine and interpret the meaning of rates of change, intercepts, zeros, extrema, and trends.
Core 1
Core 2
Using the rule for the 10 meter platform and travel
distance is 4.9t 2 , how would the rule change if the
diving took place from a 10-meter platform and the
gravity were the same as on the Moon? (Distance
fallen in meters is given by d  0.83t 2 )
Core 3
Integrated 3
Algebra 2, CPM
Algebra 2
It is not in our curriculum
SFCC – Beginning & Intermediate Algebra
A graph gives a good approximation of the number of
food stamp recipients in millions from 1994 to 1998.
What does the slope of the graph represent?
SCC – Intermediate Algebra
Basic Algebra
Given the context above:
(a) Graph this function.
(b) Interpret the slope in the context of the problem.
(c) Interpret the y-intercept in the context of the
problem.
(d) Describe the problem domain and range.
Intermediate Algebra
Interpret the meaning of the intercepts of the
polynomial found in part (a)
Algebra Concepts
See above
FUNCTIONS
p. 5
8.4 Model situations and relationships using a variety of basic functions (linear, quadratic, logarithmic, exponential, and reciprocal) and piecewise-defined functions.
High School
Community College
Algebra 2/Trigonometry
EWU
Precalculus I
Three racers ran a race as depicted below. What is the
rate of change of each, from the start to the finish?
How does the rate of change of each change?
Precalculus
Measure the dimensions of a coke can and find out the
volume of the can in cubic centimeters. Suppose the
aluminum that is used in making the top of the can
costs .05 cents per square centimeter while the sides
and bottom are made of aluminum that costs .02 cents
per square centimeter. The soda can’s tab costs 1.5
cents. Maintaining the volume of the current coke can,
create a cost function. Determine whether coke is
packaged in the most cost effective can. Plot a table of
values and show the most effective can for the price if
you think it differs from the current can design. Should
the soda company change the dimensions of its can?
Why or why not? Is there anything about the design of
the coke can that may affect the accuracy of the
measurements? Should there be extra space in the can
(how much) or should it be filled to the brim?
Needs units on axes; is it distance vs. time or speed vs.
time or both?
According to the Theory of Relativity, the length L of
an object is a function of its velocity v with respect to
an observer.
v2
, where c is the speed of light.
c2
 object change as the
How does the length of the
velocity increases? What does the object look like if
you are traveling at the speed of light? Reword so
clearer
L(v)  10 1 
c. Abstract mathematical models from word problems and interpret solutions in the context of these source problems.
Core 1
SFCC – Beginning & Intermediate Algebra
Core 2
SCC – Intermediate Algebra
Core 3
See #a above
Integrated 3
Basic Algebra
See above
Intermediate Algebra
Interpret the solutions to P( x)  0 , where P(x) is the
polynomial from part (a).
Algebra Concepts
See above
FUNCTIONS
p. 6
8.4 Model situations and relationships using a variety of basic functions (linear, quadratic, logarithmic, exponential, and reciprocal) and piecewise-defined functions.
High School
Community College
EWU
Precalculus I
Find a formula for the area of a rectangle of side
lengths x 2 and 2  x . What is the maximum area of
the rectangle and when does it occur? For what values
of x does the answer make sense?
Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
An aspirin tablet in the shape of a right circular
cylinder has a height of 1/3 cm and a radius of 1/2 cm.
The manufacturer also wishes to market the aspirin in
capsule form. The capsule is to be 3/2 cm in total
length, in the shape of a right cylinder with
hemispheres attached at both ends. Find a function that
represents the volume of the capsule. Find the radius of
the capsule so that it’s volume is equal to that of the
tablet. Include any necessary or helpful graphs
complete with explanations. Then, build informative,
scale models of the tablet and capsule.
d. Identify and justify whether a result obtained from a function model has real world significance
Core 1
SFCC – Beginning & Intermediate Algebra
Core 2
SCC – Intermediate Algebra
Core 3
Integrated 3
Basic Algebra
See above
Intermediate Algebra
Determine which of the solutions in part (c) have real
world significance and why.
Algebra Concepts
A person standing close to the edge on top of a 200foot building throws a baseball vertically upward. The
FUNCTIONS
p. 7
8.4 Model situations and relationships using a variety of basic functions (linear, quadratic, logarithmic, exponential, and reciprocal) and piecewise-defined functions.
High School
Algebra 2, CPM
Algebra 2
Algebra 2/Trigonometry
Precalculus
The effect of the Earth’s gravity diminishes as the
distance from the Earth increases. A person’s weight at
a given height above sea level is described by the
rw
function W (h) 
where r is the Earth’s radius, h
hr
is the height above sea level, and w is the person’s
weight at sea level. Construct a graph (scaled
appropriately) that depicts the weight of the backpack
as you climb (Increase in altitude). Be sure to include
and explain all relevant and implied features of the
graph.
You now work for CSI. A corpse was discovered in a
motel room at midnight and its temperature was 82˚ F.
The temperature of the room is kept constant at 68˚ F.
Two hours later, the temperature of the corpse had
dropped to 79˚ F. Given k is a constant for the object in
question, S is the surrounding temperature, t represents
the time and theta (of time) is the temperature at the
given time, Newton’s Law of Cooling states:
  t   S 

k t1  t 2    ln  1
  t 2   S 
Find the time of death to the nearest minute. Be sure to
include a graph of hours
since death as a function of body temperature.
Community College
EWU
quadratic function s (t )  16t 2  64t  200 models the
ball’s height above the ground, s(t), in feet, t seconds
after it was thrown. How many seconds does it take
until the ball hits the ground?
Precalculus I
The current population of lynx in Montana is estimated
at 1500. If the growth rate is 10% per year what was
the population
a) 10 years ago; b) 100 years ago, c) 1 million years
ago. What do these numbers represent?