# 2.1 Linear functions  (Page 10/17)

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[link] shows the input, $w,$ and output, $k,$ for a linear function $k.$ a. Fill in the missing values of the table. b. Write the linear function $k,$ round to 3 decimal places.

 $w$ –10 5.5 67.5 b $k$ 30 –26 a –44

[link] shows the input, $p,$ and output, $q,$ for a linear function $q.$ a. Fill in the missing values of the table. b. Write the linear function $k.$

 $p$ 0.5 0.8 12 b $q$ 400 700 a 1,000,000

a. $a=11,900$ ; $b=1001.1$ b. $q\left(p\right)=1000p-100$

Graph the linear function $f$ on a domain of $\left[-10,10\right]$ for the function whose slope is $\frac{1}{8}$ and y -intercept is $\frac{31}{16}$ . Label the points for the input values of $-10$ and $10.$

Graph the linear function $f$ on a domain of $\left[-0.1,0.1\right]$ for the function whose slope is 75 and y -intercept is $-22.5$ . Label the points for the input values of $-0.1$ and $0.1.$

Graph the linear function $f$ where $f\left(x\right)=ax+b$ on the same set of axes on a domain of $\left[-4,4\right]$ for the following values of $a$ and $b.$

## Extensions

Find the value of $x$ if a linear function goes through the following points and has the following slope: $\left(x,2\right),\left(-4,6\right),\text{\hspace{0.17em}}m=3$

$x=-\frac{16}{3}$

Find the value of y if a linear function goes through the following points and has the following slope: $\left(10,y\right),\left(25,100\right),\text{\hspace{0.17em}}m=-5$

Find the equation of the line that passes through the following points: and

$x=a$

Find the equation of the line that passes through the following points: and

Find the equation of the line that passes through the following points: and

$y=\frac{d}{c-a}x-\frac{ad}{c-a}$

## Real-world applications

At noon, a barista notices that she has $20 in her tip jar. If she makes an average of$0.50 from each customer, how much will she have in her tip jar if she serves $n$ more customers during her shift?

A gym membership with two personal training sessions costs $125, while gym membership with five personal training sessions costs$260. What is cost per session?

$45 per training session. A clothing business finds there is a linear relationship between the number of shirts, $n,$ it can sell and the price, $p,$ it can charge per shirt. In particular, historical data shows that 1,000 shirts can be sold at a price of $30$ , while 3,000 shirts can be sold at a price of$22. Find a linear equation in the form $p\left(n\right)=mn+b$ that gives the price $p$ they can charge for $n$ shirts.

A phone company charges for service according to the formula: $C\left(n\right)=24+0.1n,$ where $n$ is the number of minutes talked, and $C\left(n\right)$ is the monthly charge, in dollars. Find and interpret the rate of change and initial value.

The rate of change is 0.1. For every additional minute talked, the monthly charge increases by $0.1 or 10 cents. The initial value is 24. When there are no minutes talked, initially the charge is$24.

A farmer finds there is a linear relationship between the number of bean stalks, $n,$ she plants and the yield, $y,$ each plant produces. When she plants 30 stalks, each plant yields 30 oz of beans. When she plants 34 stalks, each plant produces 28 oz of beans. Find a linear relationships in the form $y=\mathrm{mn}+b$ that gives the yield when $n$ stalks are planted.

A city’s population in the year 1960 was 287,500. In 1989 the population was 275,900. Compute the rate of growth of the population and make a statement about the population rate of change in people per year.

The slope is $-400.$ This means for every year between 1960 and 1989, the population dropped by 400 per year in the city.

A town’s population has been growing linearly. In 2003, the population was 45,000, and the population has been growing by 1,700 people each year. Write an equation, $P\left(t\right),$ for the population $t$ years after 2003.

Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: $I\left(x\right)=1054x+23,286,$ where $x$ is the number of years after 1990. Which of the following interprets the slope in the context of the problem?

1. As of 1990, average annual income was $23,286. 2. In the ten-year period from 1990–1999, average annual income increased by a total of$1,054.
3. Each year in the decade of the 1990s, average annual income increased by $1,054. 4. Average annual income rose to a level of$23,286 by the end of 1999.

c.

When temperature is 0 degrees Celsius, the Fahrenheit temperature is 32. When the Celsius temperature is 100, the corresponding Fahrenheit temperature is 212. Express the Fahrenheit temperature as a linear function of $C,$ the Celsius temperature, $F\left(C\right).$

1. Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius.
2. Find and interpret $F\left(28\right).$
3. Find and interpret $F\left(–40\right).$

how can are find the domain and range of a relations
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
6000
Robert
more than 6000
Robert
can I see the picture
How would you find if a radical function is one to one?
how to understand calculus?
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
what is foci?
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
i want to sure my answer of the exercise
what is the diameter of(x-2)²+(y-3)²=25
how to solve the Identity ?
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
what is a complex number used for?
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim