# 2.3 Modeling with linear functions  (Page 6/8)

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

• 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 $\text{\hspace{0.17em}}y=0,\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}x=3,\text{\hspace{0.17em}}$ 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

6 square units

Find the area of a triangle bounded by the x -axis, the line $\text{\hspace{0.17em}}f\left(x\right)=12–\frac{1}{3}x,\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}f\left(x\right)=9–\frac{6}{7}x,\text{\hspace{0.17em}}$ and the line perpendicular to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ that passes through the origin.

20.012 square units

Find the area of a parallelogram bounded by the x -axis, the line $\text{\hspace{0.17em}}g\left(x\right)=2,\text{\hspace{0.17em}}$ the line $\text{\hspace{0.17em}}f\left(x\right)=3x,\text{\hspace{0.17em}}$ 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, $\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}$ 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.

what is set?
a colony of bacteria is growing exponentially doubling in size every 100 minutes. how much minutes will it take for the colony of bacteria to triple in size
I got 300 minutes. is it right?
Patience
no. should be about 150 minutes.
Jason
It should be 158.5 minutes.
Mr
ok, thanks
Patience
100•3=300 300=50•2^x 6=2^x x=log_2(6) =2.5849625 so, 300=50•2^2.5849625 and, so, the # of bacteria will double every (100•2.5849625) = 258.49625 minutes
Thomas
what is the importance knowing the graph of circular functions?
can get some help basic precalculus
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
can get some help inverse function
ismail
Rectangle coordinate
how to find for x
it depends on the equation
Robert
yeah, it does. why do we attempt to gain all of them one side or the other?
Melissa
whats a domain
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
Spiro; thanks for putting it out there like that, 😁
Melissa
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
difference between calculus and pre calculus?
give me an example of a problem so that I can practice answering
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
I want to learn about the law of exponent
explain this
what is functions?
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich