4.1 Linear functions  (Page 17/27)

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

$\left(a,b\right)$ and $\left(a,b+1\right)$

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

$(2a,b)$ and $(a,b+1)$

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

$(a,0)$ and $(c,d)$

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

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

Find the point of intersection of the lines $f$ and $g.$

Where is $f\left(x\right)$ greater than $g\left(x\right)?$ Where is $g\left(x\right)$ greater than $f\left(x\right)?$

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?

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 $p(n)=mn+b$ that gives the price $p$ they can charge for $n$ shirts.

A phone company charges for service according to the formula: $C(n)=24+0.1n,$ where $n$ is the number of minutes talked, and $C(n)$ 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, $n,$ 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 $y=mn+b$ 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.

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, $P(t),$ 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: $I(x)=1054x+23,286,$ 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.

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 $F(28).$
3. Find and interpret $F(\mathrm{–40}).$