4.1 Linear functions  (Page 17/27)

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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:

$\left(2a,b\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(a,b+1\right)$

$y=-\frac{1}{2}x+b+2$

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

$\left(a,0\right)$ and $\text{\hspace{0.17em}}\left(c,d\right)$

Find the equation of the line parallel to the line $\text{\hspace{0.17em}}g\left(x\right)=-0.\text{01}x\text{+2}\text{.01}\text{\hspace{0.17em}}$ through the point $\text{\hspace{0.17em}}\left(1,\text{2}\right).$

y = –0.01 x + 2.01

Find the equation of the line perpendicular to the line $\text{\hspace{0.17em}}g\left(x\right)=-0.\text{01}x\text{+2}\text{.01}\text{\hspace{0.17em}}$ through the point $\text{\hspace{0.17em}}\left(1,\text{2}\right).$

For the following exercises, use the functions

Find the point of intersection of the lines $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g.$

Where is $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ greater than $\text{\hspace{0.17em}}g\left(x\right)?\text{\hspace{0.17em}}$ Where is $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ greater than $\text{\hspace{0.17em}}f\left(x\right)?$

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 $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ more customers during her shift?

$20+0.5n$

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?

A clothing business finds there is a linear relationship between the number of shirts, $\text{\hspace{0.17em}}n,$ it can sell and the price, $\text{\hspace{0.17em}}p,$ it can charge per shirt. In particular, historical data shows that 1,000 shirts can be sold at a price of $\text{\hspace{0.17em}}30,$ while 3,000 shirts can be sold at a price of $22. Find a linear equation in the form $\text{\hspace{0.17em}}p\left(n\right)=mn+b\text{\hspace{0.17em}}$ that gives the price $\text{\hspace{0.17em}}p\text{\hspace{0.17em}}$ they can charge for $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ shirts. $p\left(n\right)=-0.004n+34$ A phone company charges for service according to the formula: $\text{\hspace{0.17em}}C\left(n\right)=24+0.1n,$ where $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is the number of minutes talked, and $\text{\hspace{0.17em}}C\left(n\right)\text{\hspace{0.17em}}$ is the monthly charge, in dollars. Find and interpret the rate of change and initial value. A farmer finds there is a linear relationship between the number of bean stalks, $\text{\hspace{0.17em}}n,$ she plants and the yield, $\text{\hspace{0.17em}}y,$ 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 $\text{\hspace{0.17em}}y=mn+b\text{\hspace{0.17em}}$ that gives the yield when $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ stalks are planted. $y=-0.5n+45$ 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. 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, $\text{\hspace{0.17em}}P\left(t\right),$ for the population $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ years after 2003. $P\left(t\right)=1700t+45,000$ Suppose that average annual income (in dollars) for the years 1990 through 1999 is given by the linear function: $\text{\hspace{0.17em}}I\left(x\right)=1054x+23,286,$ where $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ 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.

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 $\text{\hspace{0.17em}}C,$ the Celsius temperature, $\text{\hspace{0.17em}}F\left(C\right).$

1. Find the rate of change of Fahrenheit temperature for each unit change temperature of Celsius.
2. Find and interpret $\text{\hspace{0.17em}}F\left(28\right).$
3. Find and interpret $\text{\hspace{0.17em}}F\left(–40\right).$

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