Use point-by-point plotting to sketch the graph of each equation.

$$

y=x+1

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each equation.

$$

x=y+1

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each equation.

$$

x=y^{2}

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each equation.

$$

y=x^{2}

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each equation.

$$

y=x^{3}

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each equation.

$$

x=y^{3}

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each equation.

$$

x y=-6

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each equation.

$$

x y=12

$$

Wendi Z.

Numerade Educator

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.

$$

y=10-3 x

$$

Wendi Z.

Numerade Educator

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.

$$

x y-4=0

$$

Wendi Z.

Numerade Educator

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.

$$

x^{2}-y=8

$$

Wendi Z.

Numerade Educator

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.

$$

y=5 x+\frac{1}{2}(7-10 x)

$$

Wendi Z.

Numerade Educator

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.

$$

y=\frac{2+x}{3}+\frac{2-x}{3}

$$

Wendi Z.

Numerade Educator

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.

$$

3 x+4 y=5

$$

Wendi Z.

Numerade Educator

Each equation specifies a function with independent variable x. Determine whether the function is linear, constant, or neither.

$$

9 x-2 y+6=0

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each function.

$$

f(x)=1-x

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each function.

$$

f(x)=\frac{x}{2}-3

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each function.

$$

f(x)=x^{2}-1

$$

Wendi Z.

Numerade Educator

Use point-by-point plotting to sketch the graph of each function.

$$

f(x)=3-x^{2}

$$

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Use point-by-point plotting to sketch the graph of each function.

$$

f(x)=4-x^{3}

$$

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Use point-by-point plotting to sketch the graph of each function.

$$

f(x)=x^{3}-2

$$

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Use point-by-point plotting to sketch the graph of each function.

$$

f(x)=\frac{8}{x}

$$

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Use point-by-point plotting to sketch the graph of each function.

$$

f(x)=\frac{-6}{x}

$$

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The three points in the table are on the graph of the indicated function $f .$ Do these three points provide sufficient information for you to sketch the graph of $y=f(x)$ ? Add more points to the table until you are satisfied that your sketch is a good representation of the graph of $y=f(x)$ for $-5 \leq x \leq 5$.

$$

\begin{array}{lccc}

x & -1 & 0 & 1 \\

f(x) & -1 & 0 & 1

\end{array} \quad f(x)=\frac{2 x}{x^{2}+1}

$$

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The three points in the table are on the graph of the indicated function $f .$ Do these three points provide sufficient information for you to sketch the graph of $y=f(x)$ ? Add more points to the table until you are satisfied that your sketch is a good representation of the graph of $y=f(x)$ for $-5 \leq x \leq 5$.

$$

\begin{array}{lccc}

x & 0 & 1 & 2 \\

f(x) & 0 & 1 & 2

\end{array} \quad f(x)=\frac{3 x^{2}}{x^{2}+2}

$$

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Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some

problems may have more than one answer.

$$

y=f(-5)

$$

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Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some

problems may have more than one answer.

$$

y=f(4)

$$

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Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some

problems may have more than one answer.

$$

y=f(5)

$$

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Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some

problems may have more than one answer.

$$

y=f(-2)

$$

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Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some

problems may have more than one answer.

$$

0=f(x)

$$

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Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some

problems may have more than one answer.

$$

3=f(x), x<0

$$

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Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some

problems may have more than one answer.

$$

-4=f(x)

$$

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Use the following graph of a function f to determine x or y to the nearest integer, as indicated. Some

problems may have more than one answer.

$$

4=f(x)

$$

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Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.

$$

2 x+5 y=10

$$

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Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.

$$

6 x-7 y=21

$$

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Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.

$$

y(x+y)=4

$$

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Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.

$$

x(x+y)=4

$$

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Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.

$$

x^{-3}+y^{3}=27

$$

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Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.

$$

x^{2}+y^{2}=9

$$

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Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.

$$

x^{3}-y^{2}=0

$$

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Does the equation specify a function with independent variable $x$ ? If so, find the domain of the function. If not, find a value of $x$ to which there corresponds more than one value of $y$.

$$

\sqrt{x}-y^{3}=0

$$

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Find and simplify the expression if $f(x)=x^{2}-4$.

$$

f\left(x^{3}\right)

$$

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Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$

(A) $f(x+h)$

(B) $f(x+h)-f(x)$

(C) $\frac{f(x+h)-f(x)}{h}$

$$

f(x)=4 x-3

$$

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Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$

(A) $f(x+h)$

(B) $f(x+h)-f(x)$

(C) $\frac{f(x+h)-f(x)}{h}$

$$

f(x)=-3 x+9

$$

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Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$

(A) $f(x+h)$

(B) $f(x+h)-f(x)$

(C) $\frac{f(x+h)-f(x)}{h}$

$$

f(x)=4 x^{2}-7 x+6

$$

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Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$

(A) $f(x+h)$

(B) $f(x+h)-f(x)$

(C) $\frac{f(x+h)-f(x)}{h}$

$$

f(x)=3 x^{2}+5 x-8

$$

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Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$

(A) $f(x+h)$

(B) $f(x+h)-f(x)$

(C) $\frac{f(x+h)-f(x)}{h}$

$$

f(x)=x(20-x)

$$

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Find and simplify each of the following, assuming $h \neq 0$ in $(C) .$

(A) $f(x+h)$

(B) $f(x+h)-f(x)$

(C) $\frac{f(x+h)-f(x)}{h}$

$$

f(x)=x(x+40)

$$

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Refer to the area A and perimeter $P$ of a rectangle with length $l$ and width $w$ (see the figure).

The area of a rectangle is $25 \mathrm{sq}$ in. Express the perimeter $P(w)$ as a function of the width $w,$ and state the domain of this function.

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Refer to the area A and perimeter $P$ of a rectangle with length $l$ and width $w$ (see the figure).

The area of a rectangle is 81 sq in. Express the perimeter $P(l)$ as a function of the length $l,$ and state the domain of this function.

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Refer to the area A and perimeter $P$ of a rectangle with length $l$ and width $w$ (see the figure).

The perimeter of a rectangle is $100 \mathrm{~m}$. Express the area $A(l)$ as a function of the length $l,$ and state the domain of this function.

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Refer to the area A and perimeter $P$ of a rectangle with length $l$ and width $w$ (see the figure).

The perimeter of a rectangle is $160 \mathrm{~m}$. Express the area $A(w)$ as a function of the width $w,$ and state the domain of this function.

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A company manufactures memory chips for microcomputers. Its marketing research department, using statistical techniques, collected the data shown in Table 8 where $p$ is the wholesale price per chip at which $x$ million chips can be sold. Using special analytical techniques (regression analysis), an analyst produced the following price-demand function to model the data:

$$

p(x)=75-3 x \quad 1 \leq x \leq 20

$$

$$

\begin{array}{cc}

\hline x \text { (millions) } & p(\$) \\

1 & 72 \\

4 & 63 \\

9 & 48 \\

14 & 33 \\

20 & 15

\end{array}

$$

(A) Plot the data points in Table 8 , and sketch a graph of the price-demand function in the same coordinate system.

(B) What would be the estimated price per chip for a demand of 7 million chips? For a demand of 11 million chips?

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A company manufactures notebook computers. Its marketing research department, using statistical techniques, collected the data shown in Table $9,$ where $p$ is the wholesale price per computer at which $x$ thousand computers can be sold. Using special analytical techniques (regression analysis), an analyst produced the following price-demand function to model the data:

$$

p(x)=2,000-60 x \quad 1 \leq x \leq 25

$$

$$

\begin{array}{cc}

\hline \boldsymbol{x} \text { (thousands) } & p(\$) \\

1 & 1,940 \\

8 & 1,520 \\

16 & 1,040 \\

21 & 740 \\

25 & 500 \\

\hline

\end{array}

$$

(A) Plot the data points in Table 9 , and sketch a graph of the price-demand function in the same coordinate system.

(B) What would be the estimated price per computer for a demand of 11,000 computers? For a demand of 18,000 computers?

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(A) Using the price-demand function

$$

p(x)=75-3 x \quad 1 \leq x \leq 20

$$

from Problem $85,$ write the company's revenue function and indicate its domain.

(B) Complete Table 10 , computing revenues to the nearest million dollars.

(C) Plot the points from part (B) and sketch a graph of the revenue function using these points. Choose millions for the units on the horizontal and vertical axes.

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(A) Using the price-demand function

$$

p(x)=2,000-60 x \quad 1 \leq x \leq 25

$$

from Problem 86 , write the company's revenue function and indicate its domain.

(B) Complete Table 11 , computing revenues to the nearest thousand dollars.

(C) Plot the points from part (B) and sketch a graph of the revenue function using these points. Choose thousands for the units on the horizontal and vertical axes.

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The financial department for the company in Problems 85 and 87 established the following cost function for producing and selling $x$ million memory chips:

$C(x)=125+16 x$ million dollars

(A) Write a profit function for producing and selling $x$ million memory chips and indicate its domain.

(B) Complete Table 12 , computing profits to the nearest million dollars.

(C) Plot the points in part (B) and sketch a graph of the profit function using these points.

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The financial department for the company in Problems 86 and 88 established the following cost function for producing and selling $x$ thousand notebook computers:

$$

C(x)=4,000+500 x \text { thousand dollars }

$$

(A) Write a profit function for producing and selling $x$ thousand notebook computers and indicate its domain.

(B) Complete Table 13 , computing profits to the nearest thousand dollars.

(C) Plot the points in part (B) and sketch a graph of the profit function using these points.

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A candy box will be made out of a piece of cardboard that measures 8 by 12 in. Equal-sized squares $x$ inches on a side will be cut out of each corner, and then the ends and sides will be folded up to form a rectangular box.

(A) Express the volume of the box $V(x)$ in terms of $x$.

(B) What is the domain of the function $V$ (determined by the physical restrictions)?

(C) Complete Table 14 .

(D) Plot the points in part (C) and sketch a graph of the volume function using these points.

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Refer to Problem $91 .$

(A) Table 15 shows the volume of the box for some values of $x$ between 1 and 2 . Use these values to estimate to one

decimal place the value of $x$ between 1 and 2 that would produce a box with a volume of $65 \mathrm{cu}$ in.

$$

\begin{array}{cc}

\boldsymbol{x} & V(x) \\

1.1 & 62.524 \\

1.2 & 64.512 \\

1.3 & 65.988 \\

1.4 & 66.976 \\

1.5 & 67.5 \\

1.6 & 67.584 \\

1.7 & 67.252 \\

\hline

\end{array}

$$

(B) Describe how you could refine this table to estimate $x$ to two decimal places.

(C) Carry out the refinement you described in part (B) and approximate $x$ to two decimal places.

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Refer to Problems 91 and $92 .$

(A) Examine the graph of $V(x)$ from Problem $91 \mathrm{D}$ and discuss the possible locations of other values of $x$ that would produce a box with a volume of $65 \mathrm{cu}$ in.

(B) Construct a table like Table 15 to estimate any such value to one decimal place.

(C) Refine the table you constructed in part (B) to provide an approximation to two decimal places.

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A parcel delivery service will only deliver packages with length plus girth (distance around) not exceeding 108 in. A rectangular shipping box with square ends $x$ inches on a side is to be used.

(A) If the full 108 in. is to be used, express the volume of the box $V(x)$ in terms of $x$.

(B) What is the domain of the function $V$ (determined by the physical restrictions)?

(C) Complete Table 16 .

(D) Plot the points in part (C) and sketch a graph of the volume function using these points.

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In a study of the speed of muscle contraction in frogs under various loads, British biophysicist

A. W. Hill determined that the weight $w$ (in grams) placed on the muscle and the speed of contraction $v$ (in centimeters per second) are approximately related by an equation of the form

$$

(w+a)(v+b)=c

$$

where $a, b,$ and $c$ are constants. Suppose that for a certain muscle, $a=15, b=1,$ and $c=90 .$ Express $v$ as a function of $w$. Find the speed of contraction if a weight of $16 \mathrm{~g}$ is placed on the muscle.

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The percentage $s$ of seats in the House of Representatives won by Democrats and the percentage $v$ of votes cast for Democrats (when expressed as decimal fractions) are related by the equation

$$

5 v-2 s=1.4 \quad 0<s<1, \quad 0.28<v<0.68

$$

(A) Express $v$ as a function of $s$ and find the percentage of votes required for the Democrats to win $51 \%$ of the seats.

(B) Express $s$ as a function of $v$ and find the percentage of seats won if Democrats receive $51 \%$ of the votes.

Wendi Z.

Numerade Educator