Chapter 4, Linear and Quadratic Functions Video Solutions, Algebra and Trigonometry

Section 1

Properties of Linear Functions and Linear Models

01:19

Problem 1

Graph $y=3 x-1$

Mrinal R.

Numerade Educator

00:49

Problem 2

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

Tiffany T.

Numerade Educator

01:04

Problem 3

Find the average rate of change of $f(x)=-4 x+3$, from 2 to $4$.

Mrinal R.

Numerade Educator

00:57

Problem 4

Solve: $60 x-900=-15 x+2850 .$

Tiffany T.

Numerade Educator

00:16

Problem 5

If $f(x)=7.5 x+15,$ find $f(-2)$

Mrinal R.

Numerade Educator

00:40

Problem 6

The function $f(x)=\frac{2}{3} x+15$ is increasing on the interval $(-\infty, \infty)$

Mrinal R.

Numerade Educator

00:50

Problem 7

For the graph of the linear function $f(x)=m x+b, m$ is the ____ and $b$ is the _____.

Ankit G.

Numerade Educator

01:01

Problem 8

If the slope $m$ of the graph of a linear function is _____ the function is increasing over its domain.

Ankit G.

Numerade Educator

00:27

Problem 9

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

Mrinal R.

Numerade Educator

00:55

Problem 10

The average rate of change of $f(x)=2 x+8$ is 8.

Mrinal R.

Numerade Educator

02:00

Problem 11

What is the only type of function that has a constant average rate of change?
(a) linear function
(b) quadratic function
(c) step function
(d) absolute value function

Ankit G.

Numerade Educator

02:18

A car has 12,500 miles on its odometer. Say the car is driven an average of 40 miles per day. Choose the model that expresses the number of miles $N$ that will be on its odometer after $x$ days.
(a) $N(x)=-40 x+12,500$
(b) $N(x)=40 x-12,500$
(c) $N(x)=12,500 x+40$
(d) $N(x)=40 x+12,500$

Ankit G.

Numerade Educator

02:11

Problem 13

A linear function is given.
(a) Find the slope and y-intercept of each function.
(b) Use the slope and y-intercept to graph each function.
(c) What is the average rate of change of each function?
(d) Determine whether each function is increasing, decreasing, or constant.
$$
f(x)=2 x+3
$$

Mrinal R.

Numerade Educator

01:36

Problem 14

Mrinal R.

Numerade Educator

01:31

Problem 15

Mrinal R.

Numerade Educator

01:21

Problem 16

Mrinal R.

Numerade Educator

01:10

Problem 17

Mrinal R.

Numerade Educator

01:19

Problem 18

Mrinal R.

Numerade Educator

00:45

Problem 19

Mrinal R.

Numerade Educator

00:46

Problem 20

Mrinal R.

Numerade Educator

01:13

Problem 21

Determine whether each function is linear or nonlinear. If it is linear, determine the slope.
$$
\begin{array}{|rc|}
\hline \boldsymbol{x} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-2 & 4 \\
-1 & 1 \\
0 & -2 \\
1 & -5 \\
2 & -8 \\
\hline
\end{array}
$$

Ankit G.

Numerade Educator

01:06

Problem 22

Determine whether each function is linear or nonlinear. If it is linear, determine the slope.
$$
\begin{array}{|cc|}
\hline \boldsymbol{x} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-2 & 1 / 4 \\
-1 & 1 / 2 \\
0 & 1 \\
1 & 2 \\
2 & 4 \\
\hline
\end{array}
$$

Ankit G.

Numerade Educator

00:47

Problem 23

Determine whether each function is linear or nonlinear. If it is linear, determine the slope.
$$
\begin{array}{|rr|}
\hline \boldsymbol{x} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-2 & -8 \\
-1 & -3 \\
0 & 0 \\
1 & 1 \\
2 & 0 \\
\hline
\end{array}
$$

Ankit G.

Numerade Educator

01:22

Problem 24

Determine whether each function is linear or nonlinear. If it is linear, determine the slope.
$$
\begin{array}{|rc|}
\hline \boldsymbol{x} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-2 & -4 \\
-1 & 0 \\
0 & 4 \\
1 & 8 \\
2 & 12 \\
\hline
\end{array}
$$

Melissa S.

Numerade Educator

00:59

Problem 25

Determine whether each function is linear or nonlinear. If it is linear, determine the slope.
$$
\begin{array}{|rr|}
{\boldsymbol{x}} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-2 & -26 \\
-1 & -4 \\
0 & 2 \\
1 & -2 \\
2 & -10 \\
\hline
\end{array}
$$

Ankit G.

Numerade Educator

01:26

Problem 26

Determine whether each function is linear or nonlinear. If it is linear, determine the slope.
$$
\begin{array}{|rr|}
{\boldsymbol{x}} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-2 & -4 \\
-1 & -3.5 \\
0 & -3 \\
1 & -2.5 \\
2 & -2 \\
\hline
\end{array}
$$

Ankit G.

Numerade Educator

01:00

Problem 27

Determine whether each function is linear or nonlinear. If it is linear, determine the slope.
$$
\begin{array}{|rc|}
{\boldsymbol{x}} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-2 & 8 \\
-1 & 8 \\
0 & 8 \\
1 & 8 \\
2 & 8 \\
\hline
\end{array}
$$

Ankit G.

Numerade Educator

00:55

Problem 28

Determine whether each function is linear or nonlinear. If it is linear, determine the slope.
$$
\begin{array}{|rc|}
{\boldsymbol{x}} & \boldsymbol{y}=\boldsymbol{f}(\boldsymbol{x}) \\
\hline-2 & 0 \\
-1 & 1 \\
0 & 4 \\
1 & 9 \\
2 & 16 \\
\hline
\end{array}
$$

Melissa S.

Numerade Educator

04:32

Problem 29

Suppose that $f(x)=4 x-1$ and $g(x)=-2 x+5$
(a) Solve $f(x)=0$
(b) Solve $f(x)>0$
(c) Solve $f(x)=g(x)$
(d) Solve $f(x) \leq g(x)$.
(e) Graph $y=f(x)$ and $y=g(x)$ and label the point that represents the solution to the equation $f(x)=g(x)$.

Ankit G.

Numerade Educator

04:56

Problem 30

Suppose that $f(x)=3 x+5$ and $g(x)=-2 x+15$
(a) Solve $f(x)=0$.
(b) Solve $f(x)<0$.
(c) Solve $f(x)=g(x)$.
(d) Solve $f(x) \geq g(x)$
(e) Graph $y=f(x)$ and $y=g(x)$ and label the point that represents the solution to the equation $f(x)=g(x)$.

Ankit G.

Numerade Educator

03:33

Problem 31

In parts (a)-(f), use the figure below.
(a) Solve $f(x)=50$.
(b) Solve $f(x)=80$.
(c) Solve $f(x)=0$
(d) Solve $f(x)>50$.
(e) Solve $f(x) \leq 80$.
(f) Solve $0<f(x)<80$.

Ankit G.

Numerade Educator

03:32

Problem 32

In parts (a)-(f), use the figure below.
(a) Solve $g(x)=20$
(b) Solve $g(x)=60$.
(c) Solve $g(x)=0$
(d) Solve $g(x)>20$.
(e) Solve $g(x) \leq 60$.
(f) Solve $0<g(x)<60$.

Ankit G.

Numerade Educator

01:24

Problem 33

In parts (a) and (b), use the figure below.
(a) Solve the equation: $f(x)=g(x)$
(b) Solve the inequality: $f(x)>g(x)$.

Ankit G.

Numerade Educator

01:23

Problem 34

In parts (a) and (b), use the figure below.
(a) Solve the equation: $f(x)=g(x)$.
(b) Solve the inequality: $f(x) \leq g(x)$.

Ankit G.

Numerade Educator

Problem 35

In parts (a) and (b), use the figure below.
(a) Solve the equation: $f(x)=g(x)$.
(b) Solve the inequality: $g(x) \leq f(x)<h(x)$.

Check back soon!

Problem 36

In parts (a) and (b), use the figure below.
(a) Solve the equation: $f(x)=g(x)$.
(b) Solve the inequality: $g(x)<f(x) \leq h(x)$.

Check back soon!

01:49

Problem 37

The cost $C$, in dollars, to tow a car is modeled by the function $C(x)=2.5 x+85,$ where $x$ is the number of miles towed.
(a) What is the cost of towing a car 40 miles?
(b) If the cost of towing a car is $\$ 245,$ how many miles was it towed?
(c) Suppose that you have only $\$ 150 .$ What is the maximum number of miles that you can be towed?
(d) What is the domain of $C ?$

Pawan Y.

Numerade Educator

03:47

Problem 38

The monthly cost $C$, in dollars, for calls from the United States to Germany on a certain wireless plan is modeled by the function $C(x)=0.26 x+5,$ where $x$ is the number of minutes used.
(a) What is the cost if you talk on the phone for 50 minutes?
(b) Suppose that your monthly bill is $\$ 21.64 .$ How many minutes did you use the phone?
(c) Suppose that you budget $\$ 50$ per month for calls to Germany.What is the maximum number of minutes that you can talk?
(d) What is the domain of $C$ if there are 30 days in the month?

Mrinal R.

Numerade Educator

01:19

Problem 39

The relationship between the height $H$ of an adult male and the length $x$ of his humerus, in centimeters, can be modeled by the linear function $H(x)=2.89 x+78.10$.
(a) If incomplete skeletal remains of an adult male include a humerus measuring 37.1 centimeters, approximate the height of this male to the nearest tenth.
(b) If an adult male is 175.3 centimeters tall, approximate the length of his humerus to the nearest tenth.

Mrinal R.

Numerade Educator

01:17

Problem 40

The relationship between the height $H$ of an adult female and the length $x$ of her femur, in centimeters, can be modeled by the linear function $H(x)=2.47 x+54.10$.
(a) If incomplete skeletal remains of an adult female include a femur measuring 46.8 centimeters, approximate the height of this female to the nearest tenth.
(b) If an adult female is 152.4 centimeters tall, approximate the length of her femur to the nearest tenth.

Mrinal R.

Numerade Educator

06:30

Problem 41

Suppose that the quantity supplied $S$ and the quantity demanded $D$ of T-shirts at a concert are given by the following functions:
$$\begin{array}{l}S(p)=-600+50 p \\D(p)=1200-25 p\end{array}$$
where $p$ is the price of a T-shirt.
(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?

Mrinal R.

Numerade Educator

02:48

Problem 42

Suppose that the quantity supplied $S$ and the quantity demanded $D$ of hot dogs at a baseball game are given by the following functions:$$\begin{array}{l}S(p)=-2000+3000 p \\D(p)=10,000-1000 p\end{array}$$
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?

Melissa S.

Numerade Educator

05:33

Problem 43

Taxes The function $T(x)=0.12(x-9525)+952.50$ represents the tax bill $T$ of a single person whose adjusted gross income is $x$ dollars for income over $\$ 9525$ but not over $\$ 38,700,$ in 2018
(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 $\$ 3109.50 ?$

Mrinal R.

Numerade Educator

03:25

Problem 44

Under the $2017-2021$ labor agreement between Major League Baseball and the players, any team whose payroll exceeded $\$ 195$ million in 2017 had to pay a competitive balance tax of $50 \%$. The linear function $T(p)=0.50(p-195)$ describes the competitive balance $\operatorname{tax} T$ for a team whose payroll was $p$ (in millions of dollars).
(a) What is the domain of this linear function?
(b) What was the competitive balance tax for the New York Yankees, whose 2017 payroll was $\$ 209.3$ million?
(c) Graph the linear function.
(d) What was the 2017 payroll for the Los Angeles Dodgers who paid a competitive balance tax of $\$ 24.5$ million?

Mrinal R.

Numerade Educator

01:19

Problem 45

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) Solve the inequality $R(x)>C(x)$ to find the units that represent a profit for the company.
$$
\begin{array}{l}
R(x)=8 x \\
C(x)=4.5 x+17,500
\end{array}
$$

Mrinal R.

Numerade Educator

01:08

Problem 46

Mrinal R.

Numerade Educator

03:43

Problem 47

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 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 $\$ 2000 ?$

Mrinal R.

Numerade Educator

03:10

Problem 48

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 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 $\$ 72,000 ?$

Mrinal R.

Numerade Educator

04:30

Problem 49

The simplest cost function $C$ is a linear cost function, $C(x)=m x+b,$ 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 $\$ 3780 ?$

Ankit G.

Numerade Educator

05:52

Problem 50

Refer to Problem $49 .$ 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 $\$ 3780 ?$

Mrinal R.

Numerade Educator

03:45

Problem 51

According to Hooke's Law, a linear relationship exists between the distance that a spring stretches and the force stretching it. Suppose a weight of 0.5 kilograms causes a spring to stretch 2.75 centimeters and a weight of 1.2 kilograms causes the same spring to stretch 6.6 centimeters.
(a) Find a linear model that relates the distance $d$ of the stretch and the weight $w$.
(b) What stretch is caused by a weight of 2.4 kilograms?
(c) What weight causes a stretch of 19.8 centimeters?

Mrinal R.

Numerade Educator

03:14

Problem 52

The distance $d$ between the bottom of a suspended spring and a countertop is a linear function of the weight $w$ attached to the bottom of the spring. The bottom of the spring is 9 inches from the countertop whenthe attached weight is 1.5 pounds and 5 inches from the countertop when the attached weight is 2.5 pounds.
(a) Find a linear model that relates the distance $d$ from the countertop and the weight $w$.
(b) Find the distance between the bottom of the spring and the countertop if no weight is attached.
(c) What is the smallest weight that will make the bottom of the spring reach the countertop? (Ignore the thickness of the weight.)

Mrinal R.

Numerade Educator

01:35

Problem 53

Temperature Conversion The linear function $F(C)=\frac{9}{5} C+32$ converts degrees Celsius to degrees Fahrenheit, and the linear function $R(F)=F+459.67$ converts degrees Fahrenheit to degrees Rankine. Find a linear function that converts degrees Rankine to degrees Celsius.

Mrinal R.

Numerade Educator

05:06

Problem 54

The following data represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with $\$60$$\begin{array}{|cc|}\hline \text { Soda, } s & \text { Hot Dogs, } h \\\hline 20 & 0 \\15 & 3 \\10 & 6 \\5 & 9 \\\hline\end{array}$$
(a) Plot the ordered pairs $(s, h)$ in a Cartesian plane.
(b) Show that the number $h$ of hot dogs purchased is a linear function of the number $s$ of sodas purchased.
(c) Determine the linear function that describes the relation between $s$ and $h$
(d) What is the domain of the linear function?
(e) Graph the linear function in the Cartesian plane drawn in part (a).
(f) Interpret the slope.
(g) Interpret the intercepts.

Mrinal R.

Numerade Educator

02:20

Problem 55

Which functions might have the graph shown? (More than one answer is possible.)
(a) $f(x)=2 x-7$
(b) $g(x)=-3 x+4$
(c) $H(x)=5$
(d) $F(x)=3 x+4$
(e) $G(x)=\frac{1}{2} x+2$

Ankit G.

Numerade Educator

01:46

Problem 56

Which functions might have the graph shown? (More than one answer is possible.)
(a) $f(x)=3 x+1$
(b) $g(x)=-2 x+3$
(c) $H(x)=3$
(d) $F(x)=-4 x-1$
(e) $G(x)=-\frac{2}{3} x+3$

Ankit G.

Numerade Educator

02:37

Problem 57

Under what circumstances is a linear function $f(x)=m x+b$ odd? Can a linear function ever be even?

Ankit G.

Numerade Educator

01:38

Problem 58

Explain how the graph of $f(x)=m x+b$ can be used to solve $m x+b>0$.

Ankit G.

Numerade Educator

03:12

Problem 59

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
$$
\text { Graph } x^{2}-4 x+y^{2}+10 y-7=0
$$

David M.

Numerade Educator

00:39

Problem 60

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
If $f(x)=\frac{2 x+B}{x-3}$ and $f(5)=8,$ what is the value of $B ?$

Mrinal R.

Numerade Educator

01:01

Problem 61

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the average rate of change of $f(x)=3 x^{2}-5 x$ from 1 to 3 .

Mrinal R.

Numerade Educator

01:24

Problem 62

Mrinal R.

Numerade Educator

01:07

Problem 63

Complete the square for each quadratic function.
$$
f(x)=x^{2}-10 x+7
$$

Mrinal R.

Numerade Educator

01:39

Problem 64

Complete the square for each quadratic function.
$$
g(x)=3 x^{2}+15 x+13
$$

Mrinal R.

Numerade Educator

01:13

Problem 65

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find the $x$ -intercept $(\mathrm{s})$ and $y$ -intercepts(s) of the graph of $4 x^{2}+9 y=72$

Mrinal R.

Numerade Educator

01:55

Problem 66

Mrinal R.

Numerade Educator

03:48

Problem 67

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Suppose $f(x)=x^{2}-3 x+7$. Find an equation of the secant line containing the points $(-1, f(-1))$ and $(2, f(2))$

David M.

Numerade Educator

03:47

Problem 68

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Use a graphing utility to graph $f(x)=x^{3}-8 x^{2}+13 x-2$ over the interval $[-2,8] .$ Then, approximate any local maximum values and local minimum values, and determine where $f$ is increasing and where $f$ is decreasing. Round answers to two decimal places.

David M.

Numerade Educator