Deck 11: Polynomial and Rational Functions

Question 1

Is $y=(x-3)(x+3)$ a power function?

Answer 1

False

Question 2

Is $y=x^{6}$ a power function?

Answer 2

True

Question 3

The power function through the points $(1,6)$ and $(6,9)$ is $y=k x^{p}$ , where $k=$ -----------and $p=\ldots$ . Round the second answer to 3 decimal places.

Answer 3

Part A: $k=6$
Part B: $p=0.226$

Question 4

Suppose $y$ is directly proportional to $x$ . If $y=1$ when $x=5$ , what is the value of $x$ when $y$ is 2 ?

Question 5

The formula for the power function whose values are in the following table is $y=k x^{p}$ , where $k=$ --------- and $p=$ -----------

Question 6

According to one advertisement, Burger King's all-beef hamburger patties have $85 \%$ more beef than McDonald's all-beef hamburger patties. If both chains serve circular patties of the same thickness, then the diameter of Burger King's patties, $d_{B}$ , will be directly proportional to the diameter of McDonald's patties, $d_{M}$ . Which of the following formulas express $d_{B}$ as a function of $d_{M}$ ?

A) $d_{B}=d_{M}(0.85)^{1 / 2}$
B) $d_{B}=d_{M}(0.85)^{-1/ 2}$
C) $d_{B}=d_{M}(1.85)^{1 / 2}$
D) $d_{B}=d_{M}(1.85)^{-1 / 2}$

Question 7

The volume occupied by a fixed quantity of gas such as oxygen is inversely proportional to its pressure, provided that its temperature is held constant. Suppose that a quantity of oxygen occupies a 110 liter volume at a pressure of 12 atmospheres. If the temperature of the oxygen does not change, how many liters will it occupy if its pressure rises to 17 atmospheres? Round to 1 decimal place.

Question 8

When temperature is held constant, the pressure $P$ and volume $V$ of a quantity of gas are inversely proportional (Boyle's Law). The following figure shows this relationship for a particular gas. Find a formula for $V$ in terms of $P$ and use it to find $V$ when $P$ is 7 . Round to 2 decimal places.

Question 9

Poiseuille's law says that the rate at which blood is flowing through a blood vessel of radius $R$ is proportional to $R^{4}$ . For medical reasons, we want to know how a reduction in the radius of a blood vessel affects the blood flow. If the radius of the blood vessel decreases by $10 \%$ , by what percentage does the blood flow decrease? Round to the nearest whole percent.

Question 10

Which of the following graphs show $p$ being proportional to the square of $q$ ?

A)
B)
C)
D)
E)
F)

Question 11

The figure below shows the graphs of two power functions, $f$ and $g$ . The formula for $g$ is $g(x)=k x^{p}$ , where $k=$ -------- and $p=$ -------------

Question 12

The following figure gives the graphs of $f(x)=a x^{p}$ and $g(x)=b x^{q}$ .

Which is smaller, $p$ or $q$ ?

Question 13

Suppose that $a, b, c$ , and $d$ are integers and that $a$ is positive and even, $b$ is positive and odd, $c$ is negative and even, and $d$ is negative and odd. Which of the following graphs could correspond to the power function $y=a x^{c}$ ? If none of the graphs correspond to the function, enter "none".

A)
B)
C)
D)

Question 14

The following figure shows the graph of a power function, $f(x)$ , whose formula has the form $f(x)=k x^{p}$ . Which of the following statements are true? Mark all that apply.

A) $k<0$
B) $k>0$
C) $k$ must be an integer
D) $p<0$
E) $0\lt$ P $\lt1$
F) $p>1$

Question 15

The following figure gives the graphs of $f(x)=a x^{p}$ and $g(x)=b x^{q}$ .

If $g(f(1))=(a b)^{q}$ , what is $b ?$

Question 16

Let $f(x)=k x^{p}$ satisfy the conditions $f(-x)=f(x), \lim _{x \rightarrow \infty} f(x)=0$ , and $f(2)=-\frac{5}{4}$ . Which of the following must be true?

A) $p$ is odd
B) $p$ is even
C) $p$ is positive
D) $p$ is negative
E) $k$ is positive
F) $k$ is negative

Question 17

Let $f(x)=a x^{b}$ and $g(x)=c x^{d}$ be graphed in the following figure. Which of the following is true?

A) $b$ must be less than $d$
B) $b$ might be less than $d$
C) $b$ cannot be less than $d$

Question 18

Find a power function through the two points $(3,729)$ and $(6,11,664)$ .

Question 19

Is the function $y=\frac{\left(12 x^{-14}\right)\left(2 x^{5}\right)}{4 x^{-7}}$ a power function? If so, write the function in the form $y=k x^{p}$ .

Question 20

Which of the following are true?

A) $\frac{1}{8 x^{-8}}=-\frac{x^{8}}{8}$
B) $\frac{8 x^{-4}}{x^{8}}=8 x^{4}$
C) $\frac{1}{8^{8}}=8^{-8}$
D) $\frac{1}{\left(8 x^{-8}\right)^{-4}}=\frac{8^{4}}{x^{32}}$

Question 21

The functions $f$ and $F$ are given by $f(x)=k x^{p}$ and $F(x)=\sqrt{\frac{k x^{p-1}}{p}}$ .
Suppose $f(x)=\frac{x^{9}}{8}$ . What must $F$ be?

Question 22

The functions $f$ and $F$ are given by $f(x)=k x^{p}$ and $F(x)=\sqrt{\frac{k x^{p-1}}{p}}$ .
Suppose $f(x)=\frac{x^{11}}{6}$ . What must $F$ be?

A) $F=\frac{1}{\sqrt{66}} x^{5}$
B) $F=\sqrt{\frac{6}{11}} x^{5}$
C) $F=\sqrt{\frac{x^{5}}{66}}$
D) $F=\frac{1}{\sqrt{66}} x^{10}$

Question 23

Let $f(x)=2 x^{3}-4 x^{2}+5$ . Which of the following statements are true?

A) As $x \rightarrow \infty, f(x) \rightarrow \infty$
B) As $x \rightarrow \infty, f(x) \rightarrow-\infty$
C) As $x \rightarrow-\infty, f(x) \rightarrow \infty$
D) As $x \rightarrow-\infty, f(x) \rightarrow-\infty$

Question 24

Let $y=1.4 x^{5}+1.1$ . If $y$ is a polynomial, give its degree. If not, enter "not a polynomial".

Question 25

The polynomial $f$ graphed below has leading term $a x^{n}$ (i.e. $f(x)=a x^{n}+$ terms of lower degree). We know that $a$ is ----------- (positive \ negative), $n$ is ------------- (even \ odd), and the smallest possible value of $n$ is ----------------------.

Question 26

Let $f(x)=(x-2)^{2}(x+4)$ . Which of the following is the graph of $-f(x)$ ?

A)
B)
C)
D)

Question 27

Let $f(x)=(x-2)^{2}(x+4)$ . The following figure shows the graph of $f(x+a)$ , with $a$ $=$ ------------

Question 28

Let $f(x)=(x-2)^{2}(x+4)$ . Which of the following figures shows the graph of $f(2 x)$ ?

A)

B)

C)

D)

Question 29

The volume of pollutants (in millions of cubic feet) in a certain reservoir is given by $P(t)=370+25 t$ , where $t$ is time in years. The volume of the reservoir (including pollutants) is gradually increasing and is given by $R(t)=11,000+130 t$ . Let $C(t)$ be the fraction of the reservoir's volume that consists of pollutants. What percent of the reservoir's total volume consisted of pollutants in the year $t=0$ ? Round to 1 decimal place.

Question 30

The volume of pollutants (in millions of cubic feet) in a certain reservoir is given by $P(t)=400+35 t$ , where $t$ is time in years. The volume of the reservoir (including pollutants) is gradually increasing and is given by $R(t)=12,000+110 t$ . Let $C(t)$ be the fraction of the reservoir's volume that consists of pollutants. If these trends continue for many years, approximately what percent of the reservoir's total volume would eventually be pollutants? Round to 1 decimal place.

Question 31

A $12 \mathrm{~kg}$ sample of a certain alloy (mixture of metals) contains $2 \mathrm{~kg}$ of tin and $10 \mathrm{~kg}$ of copper. A chemist decides to study the properties of the alloy as its percentage of tin is varied. Suppose $x$ represents the quantity of tin, in $\mathrm{kg}$ , the chemist adds to the sample. Let $f(x)$ represent the fraction of the mixture's mass composed of tin--that is, the ratio of the tin's mass to the mixture's total mass. A negative value of $x$ represents a quantity of tin removed from the original $12 \mathrm{~kg}$ sample. What is the domain of $f$ ?

A) All real numbers $x$ .
B) All $x \geq 0$ .
C) All $x \geq 2$ .
D) All $x \geq-2$ .
E) All $x \geq-12$ .

Question 32

A $10 \mathrm{~kg}$ sample of a certain alloy (mixture of metals) contains $3 \mathrm{~kg}$ of tin and $7 \mathrm{~kg}$ of copper. A chemist decides to study the properties of the alloy as its percentage of tin is varied. Suppose $x$ represents the quantity of tin, in $\mathrm{kg}$ , the chemist adds to the sample. Let $f(x)$ represent the fraction of the mixture's mass composed of tin--that is, the ratio of the tin's mass to the mixture's total mass. A negative value of $x$ represents a quantity of tin removed from the original $10 \mathrm{~kg}$ sample. There is a zero at $x=$ ---------

Question 33

The ant population in a sandbox has been modelled by the following function: $y=343-9.5 x^{2}+102.7 x$ where $x$ is the number of months after January 2010 .
a) How many ants are in the sandbox in January 2010 ?
b) How many ants are predicted to be in the sandbox in February 2011 ?

Question 34

Find the equation of the vertical line through the $x$ -intercept of the graph $y=6(x+7)^{3}$ .

Question 35

Find the equation of the horizontal line through the $y$ -intercept of the graph $y=7 x^{3}-4 x^{2}+4$ .

Question 36

The sum of two even functions is always an odd function.

Question 37

Which of the following are polynomials:

A) $y=15 x^{4}$
B) $y=\left(x^{2}+3\right)(x-15) e^{x}$
C) $y=1-15 t+\sqrt{3 t^{5}}$
D) $y=1-15 t+\sqrt{3} t^{5}$

Question 38

Compute the following limits:
a) $\lim _{x \rightarrow \infty}\left(5 x^{3}+9 x-117\right)$
b) $\lim _{x \rightarrow-\infty}\left(5 x^{3}+9 x-117\right)$

Question 39

Compute the following limits:
a) $\lim _{x \rightarrow \infty}\left(-5 x^{4}+7 x^{3}-114 x^{2}\right)$
b) $\lim _{x \rightarrow-\infty}\left(-5 x^{4}+7 x^{3}-114 x^{2}\right)$

Question 40

Suppose that $f(x)=4 x^{5}+6 x^{3}+4 x^{2}-14$ . Select all that are true:

A) $f$ is an odd function.
B) The $y$ -intercept of $f$ is 14
C) As $x \rightarrow \infty$ then $f \rightarrow \infty$
D) As $x \rightarrow-\infty$ then $f \rightarrow \infty$

Question 41

What is the degree of the polynomial function $y=4 x^{3}-4 x+1$ ?

Question 42

Does the function $f(x)=2 x^{4}+5 x-10$ have a minimum value?

Question 43

List the zeros of the function $y=x^{3}-3 x^{2}-18 x$ in ascending order, separated by commas.

Question 44

Use a graphing calculator or computer to graph $y=x^{4}+2 x^{3}-7 x^{2}-8 x+12$ . Use the graph to pick the factored form of $y$ .

A) $y=(x-2)(x+1)(x+3)(x+2)$
B) $y=(x-2)(x-1)(x+3)(x+2)$
C) $y=(x+2)(x+1)(x-3)(x-2)$
D) $y=(x-2)(x-1)(x-3)(x+2)$

Question 45

Describe the graph of $y=\frac{1}{2}(x+2)(x-2)(x+4)$ .

A) A polynomial curve that goes down on the left and up on the right, has zeros at 4,2 , and -2 , and has a $y$ -intercept at -8 .
B) A polynomial curve that goes up on the left and down on the right, has zeros at 4, 2 , and -2 , and has a $y$ -intercept at -8 .
C) A polynomial curve that goes down on the left and up on the right, has zeros at $-4,-2$ , and 2 , and has a $y$ -intercept at -8 .
D) A polynomial curve that goes up on the left and down on the right, has zeros at $-4,-2$ , and 2 , and has a $y$ -intercept at -8 .

Question 46

Use a graphing calculator to find all the real zeros of $f(x)=0.3 x^{3}+2.7 x^{2}-0.3 x-2.7$ . List them in ascending order, separated by commas.

Question 47

The graphs of $y_{1}=0.25 x^{4}$ and $y_{2}=0.25\left(x^{4}+x^{3}-23 x^{2}+3 x+90\right)$ are shown in the figure below as viewed on the window $[-10,10]$ by $[-25,25]$ . What is the double zero of $y_{2}$ ?

Question 48

The graphs of $y_{1}=0.25 x^{4}$ and $y_{2}=0.25\left(x^{4}+x^{3}-23 x^{2}+3 x+90\right)$ are shown in the figure below as viewed on the window $[-10,10]$ by $[-25,25]$ . What happens as the viewing window is expanded?

A) The graphs become further apart from each other.
B) The graphs become indistinguishable from one another.
C) Nothing changes.

Question 49

If $f(x)=x^{2}-3$ and $g(x)=\frac{x^{4}}{2}-7$ , then $f(x)>g(x)$ on the interval ----------- (

Question 50

The function $f(x)=\sin x+0.5$ can be approximated by the function $g(x)=0.5+x-\frac{x^{3}}{6}+\frac{x^{5}}{120}$ . On what interval do the two graphs look similar?

A) $-\pi<x<0$
B) $0<x<\pi$
C) $-\frac{\pi}{2}<x<\frac{\pi}{2}$

Question 51

What are the zeros of $f(x)=x^{7}+4 x^{2}-3$ ? List them in ascending order separated by commas, and round any non-integer zeros to 3 decimal places.

Question 52

The formula for the function graphed below has leading term $a x^{n}$ (i.e. $f(x)=a x^{n}+$ terms of lower degree). If $n$ is as small as possible, then $n=$ ---------- and a= ----------

Question 53

Which of the following could represent a complete graph of $f(x)=a x+x^{3}$ where $a$ is a constant?

A)
B)
C)
D)

Question 54

A polynomial with integer coefficients having $\frac{3+\sqrt{3}}{7}$ as a zero is $f(x)=a x^{2}+b x+c$ , where $a=\ldots, b=\ldots$ , and $c=\ldots$ . Make your value for $a$ be positive and as small as possible.

Question 55

Which of the following could be a formula for the graph shown below?

A) $-\frac{1}{9}(x+3)^{2}(x-2)$
B) $(x+3)^{2}(x-2)$
C) $-\frac{1}{3}(x+3)(x-2)$
D) $-(x+3)(x-2)$

Question 56

Farmer Brown has 115 feet of fence. He wishes to close in a rectangular field using the barn as one side of the field. Suppose each side of the field perpendicular to the barn has length $x$ feet. What is the area of the fenced in field?

A) The area of the field is $x(115-2 x)$ square feet.
B) The area of the field is $x(115-x)$ square feet.
C) The area of the field is $(115-2 x)$ square feet.
D) The area of the field is $x^{2}$ square feet.

Question 57

Farmer Brown has 112 feet of fence. He wishes to close in a rectangular field using the barn as one side of the field. Suppose each side of the field perpendicular to the barn has length $x$ feet.
a) What is the area of the fenced in field in terms of $x$ ?
b) In the context of this problem, what values of $x$ make sense?
c) Approximate the maximum volume of the fenced field.

Question 58

A right circular cylinder has a volume of 122 cubic meters.
a) What is the formula for surface area in terms of the radius, $r$ , of the base of the cylinder?
b) Approximate the least possible surface area for the cylinder.
c) Approximate the radius necessary to achieve the minimum surface area.
Round answers to 3 decimal places if necessary.

Question 59

A right circular cylinder has a volume of 128 cubic meters. The least possible surface area for the cylinder is approximately:

A) 140.6008 square meters.
B) 256 square meters.
C) 296.1061 square meters.
D) 128 square meters.

Question 60

The domain of the function $y=\ln \left((2 x-3) x^{2}\right)$ is

A) all $x>0$ except for $x=\frac{3}{2}$
B) all $x>\frac{3}{2}$
C)
D) all $x>0$

Question 61

What is the domain of the function $y=\ln \left((2 x-9) x^{2}\right)$ ?

Question 62

Find the formula for a third degree polynomial with a zero at $x=-5$ , a double zero at $x=3$ , and $y$ -intercept at -135 .

Question 63

Which of the following are possible formulas for a fourth degree polynomial with at least one zero at $x=-4$ , a double zero at $x=3$ , and long-run behavior: as $x \rightarrow \infty, y \rightarrow-\infty$ . .

A) $y=-2(x+4)(x-6)(x-3)^{2}$
B) $y=2(x+4)(x-6)(x-3)^{2}$
C) $y=-(x+4)^{2}(x-3)^{2}$
D) $y=-a(x+4)(x-6)(x-3)^{2}$ where $a>0$

Question 64

What is the domain of the function $\sqrt{x^{2}-3 x-28}$ ?

Question 65

Let $f(x)=\frac{8}{x+3}$ . As $x \rightarrow-3$ from the right, $f(x) \rightarrow \ldots$ -------- .Enter "infinity" for $\infty$ and "-infinity" for $-\infty$ .

Question 66

Let $f(x)=\frac{6}{x+8}$ . As $x \rightarrow \infty, f(x) \rightarrow \ldots$ . Enter "infinity" for $\infty$ and
"-infinity" for $-\infty$ .

Question 67

Does the figure below show an accurate graph of $f(x)=\frac{27}{x-3}$ ?

Question 68

Is $\frac{e^{x}}{x^{6}+1}-\frac{x-1}{x+2}$ a rational function?

Question 69

If the function $f(x)=\frac{6}{x^{2}}+\frac{x+3}{5}$ is written in the form $f(x)=\frac{p(x)}{q(x)}$ , a ratio of polynomials, which of the following could be $p(x)$ ?

A) $x+9$
B) $6 x+18$
C) $x^{3}+9 x^{2}$
D) $x^{3}+3 x^{2}+30$

Question 70

If the function $f(x)=\frac{1}{x+3}-\frac{x}{x-5}$ is written in the form $f(x)=\frac{p(x)}{q(x)}$ , a ratio of polynomials, which of the following could be $p(x)$ ?

A) $-x^{2}-2 x-5$
B) $-x^{2}-3 x+1$
C) $1-x$
D) -8

Question 71

The horizontal asymptote of $y=\frac{2 x+1}{x^{2}}-\frac{5 x+8}{x-5}$ is $\mathrm{y}=\ldots$ . Enter "none" if there is no horizontal asymptote.

Question 72

The following figure gives the graphs of four power functions. Which one could be the graph of $y=\frac{1}{1000 x^{6}}$ ? If none of the graphs match, enter "none".

A)
B)
C)
D)
E) none

Question 73

The following figure gives the graphs of four power functions. Which one could be the graph of $y=1000 x^{-4}$ ? If none of the graphs match, enter "none".

A)
B)
C)
D)
E) none

Question 74

Must the sum of two functions with horizontal asymptotes also have a horizontal asymptote?

Question 75

A $16 \mathrm{~kg}$ sample of a certain alloy (mixture of metals) contains $5 \mathrm{~kg}$ of tin and $11 \mathrm{~kg}$ of copper. A chemist decides to study the properties of the alloy as its percentage of tin is varied. Suppose $x$ represents the quantity of tin, in $\mathrm{kg}$ , the chemist adds to the sample. Let $f(x)$ represent the fraction of the mixture's mass composed of tin--that is, the ratio of the tin's mass to the mixture's total mass. A negative value of $x$ represents a quantity of tin removed from the original $16 \mathrm{~kg}$ sample. f(0.6)= ----------- (kg tin) (kg mixture) Round to 2 decimal places.

Question 76

A $13 \mathrm{~kg}$ sample of a certain alloy (mixture of metals) contains $3 \mathrm{~kg}$ of tin and $10 \mathrm{~kg}$ of copper. A chemist decides to study the properties of the alloy as its percentage of tin is varied. Suppose $x$ represents the quantity of tin, in $\mathrm{kg}$ , the chemist adds to the sample. Let $f(x)$ represent the fraction of the mixture's mass composed of tin--that is, the ratio of the tin's mass to the mixture's total mass. A negative value of $x$ represents a quantity of tin removed from the original $13 \mathrm{~kg}$ sample. $f^{-1}(0.4)=$ ------------ kg. Round to 2 decimal places.

Question 77

The infant mortality, $I$ , in a country is related to the country's GNP (gross national product), $g$ . Some authors (Weld and Helms, 1971) have argued that the relationship is of the form $I=I_{0}+\frac{k}{g+a}$ , where $I_{0}, k$ , and $a$ are positive constants and $g \geq 0$ . For $I_{0}=3, k=8$ , and $a=4$ , the vertical asymptote of the graph of $I$ against $g$ is at $g=$ ------------ If there is no vertical asymptote, enter "none"

Question 78

Suppose that $f(x)$ is a power function and that $f(9)=\frac{1}{9}$ . Must $f(x)=\frac{1}{x}$ ?

Question 79

Which of the following are rational functions:

A) $y=\frac{x^{2}-3}{x^{4}}-\frac{1}{4 x^{2}}$
B) $y=\frac{3^{x}-4}{3^{x}}$
C) $y=\frac{3 \sqrt{x}+4}{x^{3}-4}$
D) $y=\frac{3 x^{-4}}{3+x^{4}}+\frac{x^{-3}}{1-x^{-3}}$

Question 80

Let $f(x)=\frac{2+x}{5-9 x}$ . Find $f^{-1}(x)$ .