# 4.2 Modeling with linear functions  (Page 6/9)

 Page 6 / 9

Access this online resource for additional instruction and practice with linear function models.

## Key concepts

• We can use the same problem strategies that we would use for any type of function.
• When modeling and solving a problem, identify the variables and look for key values, including the slope and y -intercept. See [link] .
• Check for reasonableness of the answer.
• Linear models may be built by identifying or calculating the slope and using the y -intercept.
• The x -intercept may be found by setting $\text{\hspace{0.17em}}y=0,$ which is setting the expression $\text{\hspace{0.17em}}mx+b\text{\hspace{0.17em}}$ equal to 0.
• The point of intersection of a system of linear equations is the point where the x - and y -values are the same. See [link] .
• A graph of the system may be used to identify the points where one line falls below (or above) the other line.

## Verbal

Explain how to find the input variable in a word problem that uses a linear function.

Determine the independent variable. This is the variable upon which the output depends.

Explain how to find the output variable in a word problem that uses a linear function.

Explain how to interpret the initial value in a word problem that uses a linear function.

To determine the initial value, find the output when the input is equal to zero.

Explain how to determine the slope in a word problem that uses a linear function.

## Algebraic

Find the area of a parallelogram bounded by the y -axis, the line $\text{\hspace{0.17em}}x=3,$ the line $\text{\hspace{0.17em}}f\left(x\right)=1+2x,$ and the line parallel to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ passing through $\text{\hspace{0.17em}}\left(\text{2},\text{7}\right).$

6 square units

Find the area of a triangle bounded by the x -axis, the line $\text{\hspace{0.17em}}f\left(x\right)=12–\frac{1}{3}x,$ and the line perpendicular to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ that passes through the origin.

Find the area of a triangle bounded by the y -axis, the line $\text{\hspace{0.17em}}f\left(x\right)=9–\frac{6}{7}x,$ and the line perpendicular to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ that passes through the origin.

20.01 square units

Find the area of a parallelogram bounded by the x -axis, the line $\text{\hspace{0.17em}}g\left(x\right)=2,$ the line $\text{\hspace{0.17em}}f\left(x\right)=3x,$ and the line parallel to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ passing through $\text{\hspace{0.17em}}\left(6,1\right).$

For the following exercises, consider this scenario: A town’s population has been decreasing at a constant rate. In 2010 the population was 5,900. By 2012 the population had dropped 4,700. Assume this trend continues.

Predict the population in 2016.

2,300

Identify the year in which the population will reach 0.

For the following exercises, consider this scenario: A town’s population has been increased at a constant rate. In 2010 the population was 46,020. By 2012 the population had increased to 52,070. Assume this trend continues.

Predict the population in 2016.

64,170

Identify the year in which the population will reach 75,000.

For the following exercises, consider this scenario: A town has an initial population of 75,000. It grows at a constant rate of 2,500 per year for 5 years.

Find the linear function that models the town’s population $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ as a function of the year, $\text{\hspace{0.17em}}t,$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is the number of years since the model began.

$P\left(t\right)=75,000+2500t$

Find a reasonable domain and range for the function $\text{\hspace{0.17em}}P.$

If the function $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ is graphed, find and interpret the x - and y -intercepts.

(–30, 0) Thirty years before the start of this model, the town had no citizens. (0, 75,000) Initially, the town had a population of 75,000.

factoring polynomial
find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has
how can we solve this problem
Sin(A+B) = sinBcosA+cosBsinA
Prove it
Eseka
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay