Linear and Quadratic Functions

Graph $y=2x-3.$

Find the slope of the line joining the points $(2,5)$ and $(-1,3).$

Find the average rate of change of $f\left(x\right)=3x^2-2$ from 2 to 4.

Solve: $60x-900=-15x+2850.$

If $f\left(x\right)=x^2-4$, find $f(-2).$

True or False The graph of the function $f\left(x\right)=x^2$ is increasing on the interval $(0,\infty ).$

For the graph of the linear function $f\left(x\right)=mx+b$ $m$ is the_______

and $b$ is the_______.

For the graph of the linear function$H\left(z\right)=-4z+3$, the slope is_____ and the y-intercept is_____.

If the slope m of the graph of a linear function is________, the function is increasing over its domain. 

True or False The slope of a nonvertical line is the average rate of change of the linear function. 

True or False If the average rate of change of a linear function is $\frac{2}{3}$, then if $y$ increases by $3$, $x$ will increase by $2$. 

True or False The average rate of change of $f\left(x\right)=2x+8$ is $8.$ 

If $f\left(x\right)=2x+3$, then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant 

If $g\left(x\right)=5x-4,$ then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant 

If $h\left(x\right)=-3x+4,$ then

a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant.

If $p\left(x\right)=-x+6,$then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant.

If $f\left(x\right)=\frac{1}{4}x-3,$then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant 

If $h\left(x\right)=-\frac{2}{3}x+4,$then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant.

If $F\left(x\right)=4,$ then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant.

If $G\left(x\right)=-2,$ then

(a) Determine the slope and y-intercept of each function.

(b) Use the slope and y-intercept to graph the linear function.

(c) Determine the average rate of change of each function.

(d) Determine whether the linear function is increasing, decreasing, or constant.

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line .

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line . 

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line.

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line . 

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line .  

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line. 

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line .  

In Problems 21–28, determine whether the given function is linear or nonlinear. If it is linear, determine the equation of the line . 

Suppose that $f\left(x\right)=4x-1$and $g\left(x\right)=-2x+5$

(a) Solve $f\left(x\right)=0$.                 (b) Solve $f\left(x\right)>0$

(c) Solve $f\left(x\right)=g\left(x\right)$             (d) Solve $f\left(x\right)\le g\left(x\right)$

(e) Graph $y=f\left(x\right)$ and $y=g\left(x\right)$ and label the point that represents the solution to the equation $f\left(x\right)=g\left(x\right)$.

Suppose that $f\left(x\right)=3x+5$and  $g\left(x\right)=-2x+15$

(a)  Solve $f\left(x\right)=0$                   (b) Solve $f\left(x\right)<0$ 

(c) Solve $f\left(x\right)=g\left(x\right)$                (d) Solve $f\left(x\right)\ge g\left(x\right)$

(e) Graph $y=f\left(x\right)$ and $y=g\left(x\right)$ and label the point that represents the solution to the equation $f\left(x\right)=g\left(x\right)$.

In parts (a) – (f), use the following figure. 

(a) Solve $f\left(x\right)=50$                        (b) Solve $f\left(x\right)=80$

(c) Solve $f\left(x\right)=0$                          (d) Solve $f\left(x\right)>50$

(e) Solve $f\left(x\right)\le 80$                        (f) Solve $0<f\left(x\right)<80$

In parts (a) and (b), use the following figure. 

(a) Solve the equation: $F\left(x\right)=g\left(x\right)$.

(b) Solve the inequality: $g\left(x\right)\le f\left(x\right)<h\left(x\right)$

In parts (a) – (f), use the following figure. 

(a) Solve $g\left(x\right)= 20$.                         (b) Solve $g\left(x\right)=60$.

(c) Solve $g\left(x\right)= 0$.                           (d) Solve $g\left(x\right)>20$.

(e) Solve $g\left(x\right)\le 60.$                         ( f ) Solve $0<g\left(x\right)<60$.

In parts (a) and (b) use the following figure

(a) Solve the equation: $f\left(x\right)=g\left(x\right)$.

(b) Solve the inequality: $f\left(x\right)>g\left(x\right)$.

In parts (a) and (b), use the following figure. 

(a) Solve the equation: $f\left(x\right)=g\left(x\right)$

(b) Solve the inequality: $f\left(x\right)\le g\left(x\right)$

In parts (a) and (b), use the following figure. 

(a) Solve the equation: $f\left(x\right)=g\left(x\right)$.

(b) Solve the inequality: $g\left(x\right)<f\left(x\right)\le h\left(x\right)$

Car Rentals The cost C, in dollars, of renting a moving truck for a day is modeled by the function $C\left(x\right)=0.25x+35$, where x is the number of miles driven.

(a) What is the cost if you drive $x=40$ miles?

(b) If the cost of renting the moving truck is \(80, how many miles did you drive? 

(c) Suppose that you want the cost to be no more than \)100. What is the maximum number of miles that you can drive? 

(d) What is the implied domain of C ? 

(e) Interpret the slope. 

( f ) Interpret the y-intercept. 

Phone Charges The monthly cost C, in dollars, for international calls on a certain cellular phone plan is modeled by the function $C\left(x\right)=0.38x+5$, where x is the number of minutes used.

 (a) What is the cost if you talk on the phone for$x = 50$ minutes?

 (b) Suppose that your monthly bill is \(29.32. How many minutes did you use the phone? 

(c) Suppose that you budget yourself \)60 per month for the phone. What is the maximum number of minutes that you can talk? 

(d) What is the implied domain of C if there are 30 days in the month? 

(e) Interpret the slope. 

( f) Interpret the y-intercept. 

Supply and Demand Suppose that the quantity supplied S and quantity demanded D of T-shirts at a concert are given by the following functions:

$S\left(p\right)=-200+50p$

$D\left(p\right)=1000-25p$

(a) Find the equilibrium price for T-shirts at this concert. What is the equilibrium quantity? 

(b) Determine the prices for which quantity demanded is greater than quantity supplied. 

(c) What do you think will eventually happen to the price of T-shirts if quantity demanded is greater than quantity supplied? 

Supply and Demand Suppose that the quantity supplied S and quantity demanded D of hot dogs at a baseball game are given by the following functions:

 $S\left(p\right)=-2000+3000p$

$D\left(p\right)=10,000-1000p$

where p is the price of a hot dog. 

(a) Find the equilibrium price for hot dogs at the baseball game. What is the equilibrium quantity? 

(b) Determine the prices for which quantity demanded is less than quantity supplied. 

(c) What do you think will eventually happen to the price of hot dogs if quantity demanded is less than quantity supplied? 

Taxes The function $T\left(x\right) = 0.15(x - 8350) + 835$ represents the tax bill T of a single person whose adjusted gross income is x dollars for income between \(8350 and \)33,950, inclusive, in 2009. 

Source: Internal Revenue Service 

(a) What is the domain of this linear function? 

(b) What is a single filer’s tax bill if adjusted gross income is \(20,000? 

(c) Which variable is independent and which is dependent? 

(d) Graph the linear function over the domain specified in part (a). (e) What is a single filer’s adjusted gross income if the tax bill is \)3707.50? 

(f) Interpret the slope. 

Luxury Tax In 2002, major league baseball signed a labor agreement with the players. In this agreement, any team whose payroll exceeded \(136.5 million in 2006 had to pay a luxury tax of 40% (for second offenses). The linear function $T\left(p\right)= 0.40\left(p - 136.5\right)$ describes the luxury tax T of a team whose payroll was p (in millions of dollars). 

Source: Major League Baseball 

(a) What is the implied domain of this linear function? 

(b) What was the luxury tax for the New York Yankees whose 2006 payroll was \)171.1 million? 

(c) Graph the linear function. 

(d) What is the payroll of a team that pays a luxury tax of $11.7 million? 

(e) Interpret the slope. 

The point at which a company’s profits equal zero is called the company’s break-even point. For Problems 43 and 44, let R represent a company’s revenue, let C represent the company’s costs, and let x represent the number of units produced and sold each day.

(a) Find the firm’s break-even point; that is, find x so that $R=C$.

 (b) Find the values of x such that$R\left(x\right)>C\left(x\right)$. This represents the number of units that the company must sell to earn a profit .

$$\begin{gathered} R\left(x\right)=8x \\ C\left(x\right)=4.5x+17500 \end{gathered}$$

The point at which a company’s profits equal zero is called the company’s break-even point. For Problems 43 and 44, let R represent a company’s revenue, let C represent the company’s costs, and let x represent the number of units produced and sold each day. 

(a) Find the firm’s break-even point; that is, find x so that $R=C$.

(b) Find the values of x such that $R\left(x\right)>C\left(x\right)$. This represents the number of units that the company must sell to earn a profit.

$$\begin{gathered} R\left(x\right)=12x \\ C\left(x\right)=10x+15000 \end{gathered}$$.

Straight-line Depreciation Suppose that a company has just purchased a new computer for \(3000. The company chooses to depreciate the computer using the straight-line method over 3 years.

(a) Write a linear model that expresses the book value V of the computer as a function of its age x. 

(b) What is the implied domain of the function found in part (a)? 

(c) Graph the linear function. 

(d) What is the book value of the computer after 2 years? 

(e) When will the computer have a book value of \)2000? 

Straight-line Depreciation Suppose that a company has just purchased a new machine for its manufacturing facility for \(120,000. The company chooses to depreciate the machine using the straight-line method over 10 years. 

(a) Write a linear model that expresses the book value V of the machine as a function of its age x. 

(b) What is the implied domain of the function found in part (a)? 

(c) Graph the linear function. 

(d) What is the book value of the machine after 4 years? 

(e) When will the machine have a book value of \)72,000? 

Cost Function The simplest cost function is the linear cost function, $C\left(x\right)=mx+b$, where the y-intercept b represents the fixed costs of operating a business and the slope m represents the cost of each item produced. Suppose that a small bicycle manufacturer has daily fixed costs of \(1800 and each bicycle costs \)90 to manufacture.

(a) Write a linear model that expresses the cost C of manufacturing x bicycles in a day. 

(b) Graph the model. 

(c) What is the cost of manufacturing 14 bicycles in a day? 

(d) How many bicycles could be manufactured for $3780? 

Cost Function Refer to Problem 47. Suppose that the landlord of the building increases the bicycle manufacturer’s rent by \(100 per month. 

(a) Assuming that the manufacturer is open for business 20 days per month, what are the new daily fixed costs? 

(b) Write a linear model that expresses the cost C of manufacturing x bicycles in a day with the higher rent. 

(c) Graph the model. 

(d) What is the cost of manufacturing 14 bicycles in a day? 

(e) How many bicycles can be manufactured for \)3780? 

Truck Rentals A truck rental company rents a truck for one day by charging \(29 plus \)0.07 per mile. 

(a) Write a linear model that relates the cost C, in dollars, of renting the truck to the number x of miles driven. 

(b) What is the cost of renting the truck if the truck is driven 110 miles? 230 miles? 

Long Distance A phone company offers a domestic long distance package by charging \(5 plus \)0.05 per minute. 

(a) Write a linear model that relates the cost C, in dollars, of talking x minutes. 

(b) What is the cost of talking 105 minutes? 180 minutes?